🕷️ Crawler Inspector

URL Lookup

Direct Parameter Lookup

Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 166 (from laksa192)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa166.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions","crawl_time":1775749667,"first_indexed_time":1635417973,"http_code":200,"src_unparsed":"com,wolframalpha!www,\/examples\/pro-features\/step-by-step-solutions s443","src_root_hash":"10472823281220336366","history_drop_reason":null,"meta_title":"Wolfram|Alpha Examples: Step-by-Step Solutions","meta_descriptions":["Step-by-step solutions for math, chemistry, physics. Calculus, algebra, trigonometry, geometry, number theory, base conversions, statistics. Physics formulas, chemical structures, chemical conversions and reactions. Intercepts, discontinuities, differential equations, linear algebra, mathematical induction."],"attrs_boilerpipe_text":"Pro\nExamples for\nSolutions for over 200 topics within math, chemistry and physics problems across all levels of education, from elementary school to college.\nArithmetic\n›\nSee how to show your work for arithmetic problems.\nSee step-by-step solutions for doing basic multiplication or division problems:\n3 × 5\n60431 \/ 89\nSee the steps for applying the order of operations for arithmetic:\n9 (3+1) + 17 \/ (6-12)\nFollow the steps to convert an improper fraction to a mixed number:\n87\/5 as a mixed number\nMore examples\nStatistics\n›\nSee the steps for analyzing data and applying statistical methods.\nPerform basic statistical analysis on a set of data, seeing the steps along the way:\naverage of {1, 2, 3, 4, 5, 6, 7}\nmode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20}\nvariance 4, 6, 7, 9, 10, 11\nquartile of {1, 2, 3, 4, 5, 6}\nMore examples\nDifferential Equations\n›\nExplore step-by-step solutions for solving equations involving derivatives.\nSee how ordinary differential equations are solved:\nsolve y' = y^2 x\ny'(t) - 2y(t) = 3 e^(2t)\nt y(t) (1 + t y(t)^2) y'(t) = 1\ny''+ y = sin(2x)\nMore examples\nProofs\n›\nGet step-by-step results for proving mathematical statements.\nSee the steps toward proving a trigonometric identity:\n(1 + tan(x))\/(1 - tan(x)) = (cos(x) + sin(x))\/(cos(x) - sin(x))\nProve a sum identity by induction:\nprove by induction sum of j from 1 to n = n(n+1)\/2 for n>0\nProve an inequality by induction:\nprove by induction (3n)! > 3^n (n!)^3 for n>0\nInvestigate the convergence or divergence of an infinite sum step by step:\nDoes the sum of 5*3^(1 - n) converge?\nMore examples\nGeometry\n›\nExplore two- and three-dimensional shapes and their properties.\nCompute properties of geometric objects step by step:\narea of a circle with radius 2\nfind the surface area of a sphere with radius = 8\nLearn how to compute different properties for graphical objects:\n(2, -4, 7) to (-4, 6, 1) distance\nexpress x^2+y^2=3 in polar coordinates\nDetermine the equation of a line with certain properties using a selected form:\nequation of line through (2, 5), (4, 1)\nMore examples\nCalculus\n›\nGet step-by-step solutions for common calculus problems, like computing limits, derivatives and integrals.\nFind the derivative using the product rule, chain rule and other methods:\ncompute the derivative of x*sin[x]\nCalculate an indefinite integral by substitution, integration by parts and other methods:\nintegrate sin(x)cos(x)^2\nLearn how to take a limit:\nlimit of (x - 3) \/ (x^2 - 2x - 3) as x approaches 3\nExplore step by step a variety of tests to discover local and global extrema:\nfind the extrema of 3x^4 - 4x^3 - 12x^2 + 1\nMore examples\nLinear Algebra\n›\nSolve problems involving vectors and matrices step by step.\nCompute properties of a matrix step by step:\n{{2, 2^2, 8\/4}, {1, - 4, 7 - 4}}\nFind the determinant step by step with various methods:\ndeterminant of {{1,2,1}, {1,1,0}, {0,1,1}}\nCompute eigenvalues and eigenvectors step by step:\neigenvalues {{3,-1},{0,2}}\nCompute a cross product step by step:\n(1,2,3)x(3,4,5)\nMore examples\nPhysics\n›\nSee how to solve physics problems step by step.\nDo physics calculations step by step:\nDoppler shift for sound at source frequency 100 kHz for 70 mph\ncoulomb's law for 0.1 C and 0.2 C at 2 meters distance\nlength contraction for a proper length of 1 yard at 0.93c\nphoton energy 435nm\nwater flow through a pipe 2\" diameter, 10' length of pipe with 2 kPa pressure difference\nMore examples\nGo\nPro\nNow\nLearn more about Wolfram|Alpha Pro »","attrs_markdown":"![no-js-running-man-logo](https:\/\/www.wolframalpha.com\/_next\/static\/images\/running-man_d23cee5b24bc562c49a0355e9408ad8c.png)Uh oh! Wolfram\\|Alpha doesn't run without JavaScript.\n\nPlease enable JavaScript. If you don't know how, you can find instructions[here](https:\/\/enable-javascript.com\/).Once you've done that, refresh this page to start using Wolfram\\|Alpha.\n#\n[All Examples›](https:\/\/www.wolframalpha.com\/examples)[Pro Features›](https:\/\/www.wolframalpha.com\/examples\/pro-features)\n\nBrowse Examples\n\nPro\n\nExamples for\n\n# Step-by-Step Solutions\nSolutions for over 200 topics within math, chemistry and physics problems across all levels of education, from elementary school to college.\n\n![Step-by-Step Solutions](https:\/\/www.wolframcdn.com\/examples\/proPages\/subpageImages\/step-by-step-solutions-hero.png)\n\n[Arithmetic ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-arithmetic)\n\nSee how to show your work for arithmetic problems.\n\n#### See step-by-step solutions for doing basic multiplication or division problems:\n[3 × 5](https:\/\/www.wolframalpha.com\/input?i=3+%C3%97+5)\n\n[60431 \/ 89](https:\/\/www.wolframalpha.com\/input?i=60431+%2F+89)\n\n#### See the steps for applying the order of operations for arithmetic:\n[9 (3+1) + 17 \/ (6-12)](https:\/\/www.wolframalpha.com\/input?i=9+%283%2B1%29+%2B+17+%2F+%286-12%29)\n\n#### Follow the steps to convert an improper fraction to a mixed number:\n[87\/5 as a mixed number](https:\/\/www.wolframalpha.com\/input?i=87%2F5+as+a+mixed+number)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-arithmetic)\n\n[Statistics ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-statistics)\n\nSee the steps for analyzing data and applying statistical methods.\n\n#### Perform basic statistical analysis on a set of data, seeing the steps along the way:\n[average of {1, 2, 3, 4, 5, 6, 7}](https:\/\/www.wolframalpha.com\/input?i=average+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%2C+7%7D)\n\n[mode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20}](https:\/\/www.wolframalpha.com\/input?i=mode+of+%7B1%2C+5%2C+4%2C+2%2C+3%2C+4%2C+5%2C+11%2C+4%2C+11%2C+20%7D)\n\n[variance 4, 6, 7, 9, 10, 11](https:\/\/www.wolframalpha.com\/input?i=variance+4%2C+6%2C+7%2C+9%2C+10%2C+11)\n\n[quartile of {1, 2, 3, 4, 5, 6}](https:\/\/www.wolframalpha.com\/input?i=quartile+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%7D)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-statistics)\n\n[Differential Equations ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-differential-equations)\n\nExplore step-by-step solutions for solving equations involving derivatives.\n\n#### See how ordinary differential equations are solved:\n[solve y' = y^2 x](https:\/\/www.wolframalpha.com\/input?i=solve+y%27+%3D+y%5E2+x)\n\n[y'(t) - 2y(t) = 3 e^(2t)](https:\/\/www.wolframalpha.com\/input?i=y%27%28t%29+-+2y%28t%29+%3D+3+e%5E%282t%29)\n\n[t y(t) (1 + t y(t)^2) y'(t) = 1](https:\/\/www.wolframalpha.com\/input?i=t+y%28t%29+%281+%2B+t+y%28t%29%5E2%29+y%27%28t%29+%3D+1)\n\n[y''+ y = sin(2x)](https:\/\/www.wolframalpha.com\/input?i=y%27%27%2B+y+%3D+sin%282x%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-differential-equations)\n\n[Proofs ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-proofs)\n\nGet step-by-step results for proving mathematical statements.\n\n#### See the steps toward proving a trigonometric identity:\n[(1 + tan(x))\/(1 - tan(x)) = (cos(x) + sin(x))\/(cos(x) - sin(x))](https:\/\/www.wolframalpha.com\/input?i=%281+%2B+tan%28x%29%29%2F%281+-+tan%28x%29%29+%3D+%28cos%28x%29+%2B+sin%28x%29%29%2F%28cos%28x%29+-+sin%28x%29%29)\n\n#### Prove a sum identity by induction:\n[prove by induction sum of j from 1 to n = n(n+1)\/2 for n\\>0](https:\/\/www.wolframalpha.com\/input?i=prove+by+induction+sum+of+j+from+1+to+n+%3D+n%28n%2B1%29%2F2+for+n%3E0)\n\n#### Prove an inequality by induction:\n[prove by induction (3n)! \\> 3^n (n!)^3 for n\\>0](https:\/\/www.wolframalpha.com\/input?i=prove+by+induction+%283n%29%21+%3E+3%5En+%28n%21%29%5E3+for+n%3E0)\n\n#### Investigate the convergence or divergence of an infinite sum step by step:\n[Does the sum of 5\\*3^(1 - n) converge?](https:\/\/www.wolframalpha.com\/input?i=Does+the+sum+of+5*3%5E%281+-+n%29+converge%3F)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-proofs)\n\n[Algebra ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-algebra)\n\nLearn how to manipulate and solve expressions with variables step by step.\n\n#### Solve equations one step at a time:\n[find t for t^2 + 3t + 2 = 0](https:\/\/www.wolframalpha.com\/input?i=find+t+for+t%5E2+%2B+3t+%2B+2+%3D+0)\n\n#### Factor polynomials step by step:\n[factor x^3+2x^2+x+2](https:\/\/www.wolframalpha.com\/input?i=factor+x%5E3%2B2x%5E2%2Bx%2B2)\n\n#### Expand polynomials using the distributive law, the binomial theorem and other procedures:\n[expanded form of (x + 3)^2](https:\/\/www.wolframalpha.com\/input?i=expanded+form+of+%28x+%2B+3%29%5E2)\n\n#### Learn to rewrite a rational expression using partial fraction decomposition:\n[partial fraction decomposition 1\/(x^2 + 4x + 3)](https:\/\/www.wolframalpha.com\/input?i=partial+fraction+decomposition+1%2F%28x%5E2+%2B+4x+%2B+3%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-algebra)\n\n[Trigonometry ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-trigonometry)\n\nLearn the steps for evaluating or manipulating trigonometric expressions.\n\n#### Show your work for solving the unknown measurements of a triangle:\n[right triangle with hypotenuse 2, A = 60 degrees](https:\/\/www.wolframalpha.com\/input?i=right+triangle+with+hypotenuse+2%2C+A+%3D+60+degrees)\n\n#### See the steps for how to find the values of a trigonometric function:\n[sin(2pi\/3)](https:\/\/www.wolframalpha.com\/input?i=sin%282pi%2F3%29)\n\n#### Get step-by-step solutions for applying the sum and difference formulas:\n[simplify sin(x-y) + sin(x+y)](https:\/\/www.wolframalpha.com\/input?i=simplify+sin%28x-y%29+%2B+sin%28x%2By%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-trigonometry)\n\n[Discrete Mathematics ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-discrete-mathematics)\n\nSee the steps involved for determining different properties of numbers.\n\n#### Find step-by-step solutions for prime factorization, primality testing, GCD and more:\n[is 73 prime?](https:\/\/www.wolframalpha.com\/input?i=is+73+prime%3F)\n\n[lcm(90, 342)](https:\/\/www.wolframalpha.com\/input?i=lcm%2890%2C+342%29)\n\n[are 12 and 21 coprime?](https:\/\/www.wolframalpha.com\/input?i=are+12+and+21+coprime%3F)\n\n#### Follow the steps to convert between bases:\n[convert 33ab3653 base 12 to base 10](https:\/\/www.wolframalpha.com\/input?i=convert+33ab3653+base+12+to+base+10)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-discrete-mathematics)\n\n[Chemistry ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-chemistry)\n\nExplore a wide array of chemistry topics with step-by-step solutions.\n\n#### Get a step-by-step procedure to draw the Lewis structures of molecules:\n[What is the Lewis structure of NO2](https:\/\/www.wolframalpha.com\/input?i=What+is+the+Lewis+structure+of+NO2)\n\n#### Follow the steps for unit conversion:\n[convert 5mL benzene to moles](https:\/\/www.wolframalpha.com\/input?i=convert+5mL+benzene+to+moles)\n\n#### Learn how to balance a chemical reaction:\n[glucose + oxygen -\\> water + carbon dioxide](https:\/\/www.wolframalpha.com\/input?i=glucose+%2B+oxygen+-%3E+water+%2B+carbon+dioxide)\n\n#### Compute oxidation states of chemicals:\n[oxidation states of chromium(III) chloride](https:\/\/www.wolframalpha.com\/input?i=oxidation+states+of+chromium%28III%29+chloride)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-chemistry)\n\n[Geometry ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-geometry)\n\nExplore two- and three-dimensional shapes and their properties.\n\n#### Compute properties of geometric objects step by step:\n[area of a circle with radius 2](https:\/\/www.wolframalpha.com\/input?i=area+of+a+circle+with+radius+2)\n\n[find the surface area of a sphere with radius = 8](https:\/\/www.wolframalpha.com\/input?i=find+the+surface+area+of+a+sphere+with+radius+%3D+8)\n\n#### Learn how to compute different properties for graphical objects:\n[(2, -4, 7) to (-4, 6, 1) distance](https:\/\/www.wolframalpha.com\/input?i=%282%2C+-4%2C+7%29+to+%28-4%2C+6%2C+1%29+distance)\n\n[express x^2+y^2=3 in polar coordinates](https:\/\/www.wolframalpha.com\/input?i=express+x%5E2%2By%5E2%3D3+in+polar+coordinates)\n\n#### Determine the equation of a line with certain properties using a selected form:\n[equation of line through (2, 5), (4, 1)](https:\/\/www.wolframalpha.com\/input?i=equation+of+line+through+%282%2C+5%29%2C+%284%2C+1%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-geometry)\n\n[Calculus ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-calculus)\n\nGet step-by-step solutions for common calculus problems, like computing limits, derivatives and integrals.\n\n#### Find the derivative using the product rule, chain rule and other methods:\n[compute the derivative of x\\*sin\\[x\\]](https:\/\/www.wolframalpha.com\/input?i=compute+the+derivative+of+x*sin%5Bx%5D)\n\n#### Calculate an indefinite integral by substitution, integration by parts and other methods:\n[integrate sin(x)cos(x)^2](https:\/\/www.wolframalpha.com\/input?i=integrate+sin%28x%29cos%28x%29%5E2)\n\n#### Learn how to take a limit:\n[limit of (x - 3) \/ (x^2 - 2x - 3) as x approaches 3](https:\/\/www.wolframalpha.com\/input?i=limit+of+%28x+-+3%29+%2F+%28x%5E2+-+2x+-+3%29+as+x+approaches+3)\n\n#### Explore step by step a variety of tests to discover local and global extrema:\n[find the extrema of 3x^4 - 4x^3 - 12x^2 + 1](https:\/\/www.wolframalpha.com\/input?i=find+the+extrema+of+3x%5E4+-+4x%5E3+-+12x%5E2+%2B+1)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-calculus)\n\n[Linear Algebra ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-linear-algebra)\n\nSolve problems involving vectors and matrices step by step.\n\n#### Compute properties of a matrix step by step:\n[{{2, 2^2, 8\/4}, {1, - 4, 7 - 4}}](https:\/\/www.wolframalpha.com\/input?i=%7B%7B2%2C+2%5E2%2C+8%2F4%7D%2C+%7B1%2C+-+4%2C+7+-+4%7D%7D)\n\n#### Find the determinant step by step with various methods:\n[determinant of {{1,2,1}, {1,1,0}, {0,1,1}}](https:\/\/www.wolframalpha.com\/input?i=determinant+of+%7B%7B1%2C2%2C1%7D%2C+%7B1%2C1%2C0%7D%2C+%7B0%2C1%2C1%7D%7D)\n\n#### Compute eigenvalues and eigenvectors step by step:\n[eigenvalues {{3,-1},{0,2}}](https:\/\/www.wolframalpha.com\/input?i=eigenvalues+%7B%7B3%2C-1%7D%2C%7B0%2C2%7D%7D)\n\n#### Compute a cross product step by step:\n[(1,2,3)x(3,4,5)](https:\/\/www.wolframalpha.com\/input?i=%281%2C2%2C3%29x%283%2C4%2C5%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-linear-algebra)\n\n[Physics ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-physics)\n\nSee how to solve physics problems step by step.\n\n#### Do physics calculations step by step:\n[Doppler shift for sound at source frequency 100 kHz for 70 mph](https:\/\/www.wolframalpha.com\/input?i=Doppler+shift+for+sound+at+source+frequency+100+kHz+for+70+mph)\n\n[coulomb's law for 0.1 C and 0.2 C at 2 meters distance](https:\/\/www.wolframalpha.com\/input?i=coulomb%27s+law+for+0.1+C+and+0.2+C+at+2+meters+distance)\n\n[length contraction for a proper length of 1 yard at 0.93c](https:\/\/www.wolframalpha.com\/input?i=length+contraction+for+a+proper+length+of+1+yard+at+0.93c)\n\n[photon energy 435nm](https:\/\/www.wolframalpha.com\/input?i=photon+energy+435nm)\n\n[water flow through a pipe 2\" diameter, 10' length of pipe with 2 kPa pressure difference](https:\/\/www.wolframalpha.com\/input?i=water+flow+through+a+pipe+2%22+diameter%2C+10%27+length+of+pipe+with+2+kPa+pressure+difference)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-physics)\n\n[GoProNow](https:\/\/www.wolframalpha.com\/pro\/pricing\/students)[Learn more about Wolfram\\|Alpha Pro »](https:\/\/www.wolframalpha.com\/pro)\n\n- [Pro](https:\/\/www.wolframalpha.com\/pro)\n- [Mobile Apps](https:\/\/products.wolframalpha.com\/mobile)\n- [Products](https:\/\/products.wolframalpha.com\/)\n- [Business](https:\/\/products.wolframalpha.com\/enterprise)\n- [API & Developer Solutions](https:\/\/products.wolframalpha.com\/api)\n- [LLM Solutions](https:\/\/www.wolfram.com\/resources\/tools-for-AIs\/)\n\n- [Resources & Tools](https:\/\/www.wolframalpha.com\/resources)\n- [About](https:\/\/www.wolframalpha.com\/about)\n- [Contact](https:\/\/www.wolframalpha.com\/contact-us)\n- [Connect](https:\/\/www.wolframalpha.com\/social)\n\n- ©2026 Wolfram\n- [Terms](https:\/\/www.wolframalpha.com\/termsofuse)\n- [Privacy](https:\/\/www.wolfram.com\/legal\/privacy\/wolfram)\n\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n- [Wolfram Language](https:\/\/www.wolfram.com\/language)\n- [Mathematica](https:\/\/www.wolfram.com\/mathematica)\n- [Wolfram Demonstrations](https:\/\/demonstrations.wolfram.com\/)\n- [Wolfram for Education](https:\/\/www.wolfram.com\/education)\n- [MathWorld](https:\/\/mathworld.wolfram.com\/)","attrs_readable_markdown":"##\nPro\n\nExamples for\n\nSolutions for over 200 topics within math, chemistry and physics problems across all levels of education, from elementary school to college.\n\n![Step-by-Step Solutions](https:\/\/www.wolframcdn.com\/examples\/proPages\/subpageImages\/step-by-step-solutions-hero.png)\n\n[Arithmetic ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-arithmetic)\n\nSee how to show your work for arithmetic problems.\n\n#### See step-by-step solutions for doing basic multiplication or division problems:\n[3 × 5](https:\/\/www.wolframalpha.com\/input?i=3+%C3%97+5)\n\n[60431 \/ 89](https:\/\/www.wolframalpha.com\/input?i=60431+%2F+89)\n\n#### See the steps for applying the order of operations for arithmetic:\n[9 (3+1) + 17 \/ (6-12)](https:\/\/www.wolframalpha.com\/input?i=9+%283%2B1%29+%2B+17+%2F+%286-12%29)\n\n#### Follow the steps to convert an improper fraction to a mixed number:\n[87\/5 as a mixed number](https:\/\/www.wolframalpha.com\/input?i=87%2F5+as+a+mixed+number)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-arithmetic)\n\n[Statistics ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-statistics)\n\nSee the steps for analyzing data and applying statistical methods.\n\n#### Perform basic statistical analysis on a set of data, seeing the steps along the way:\n[average of {1, 2, 3, 4, 5, 6, 7}](https:\/\/www.wolframalpha.com\/input?i=average+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%2C+7%7D)\n\n[mode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20}](https:\/\/www.wolframalpha.com\/input?i=mode+of+%7B1%2C+5%2C+4%2C+2%2C+3%2C+4%2C+5%2C+11%2C+4%2C+11%2C+20%7D)\n\n[variance 4, 6, 7, 9, 10, 11](https:\/\/www.wolframalpha.com\/input?i=variance+4%2C+6%2C+7%2C+9%2C+10%2C+11)\n\n[quartile of {1, 2, 3, 4, 5, 6}](https:\/\/www.wolframalpha.com\/input?i=quartile+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%7D)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-statistics)\n\n[Differential Equations ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-differential-equations)\n\nExplore step-by-step solutions for solving equations involving derivatives.\n\n#### See how ordinary differential equations are solved:\n[solve y' = y^2 x](https:\/\/www.wolframalpha.com\/input?i=solve+y%27+%3D+y%5E2+x)\n\n[y'(t) - 2y(t) = 3 e^(2t)](https:\/\/www.wolframalpha.com\/input?i=y%27%28t%29+-+2y%28t%29+%3D+3+e%5E%282t%29)\n\n[t y(t) (1 + t y(t)^2) y'(t) = 1](https:\/\/www.wolframalpha.com\/input?i=t+y%28t%29+%281+%2B+t+y%28t%29%5E2%29+y%27%28t%29+%3D+1)\n\n[y''+ y = sin(2x)](https:\/\/www.wolframalpha.com\/input?i=y%27%27%2B+y+%3D+sin%282x%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-differential-equations)\n\n[Proofs ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-proofs)\n\nGet step-by-step results for proving mathematical statements.\n\n#### See the steps toward proving a trigonometric identity:\n[(1 + tan(x))\/(1 - tan(x)) = (cos(x) + sin(x))\/(cos(x) - sin(x))](https:\/\/www.wolframalpha.com\/input?i=%281+%2B+tan%28x%29%29%2F%281+-+tan%28x%29%29+%3D+%28cos%28x%29+%2B+sin%28x%29%29%2F%28cos%28x%29+-+sin%28x%29%29)\n\n#### Prove a sum identity by induction:\n[prove by induction sum of j from 1 to n = n(n+1)\/2 for n\\>0](https:\/\/www.wolframalpha.com\/input?i=prove+by+induction+sum+of+j+from+1+to+n+%3D+n%28n%2B1%29%2F2+for+n%3E0)\n\n#### Prove an inequality by induction:\n[prove by induction (3n)! \\> 3^n (n!)^3 for n\\>0](https:\/\/www.wolframalpha.com\/input?i=prove+by+induction+%283n%29%21+%3E+3%5En+%28n%21%29%5E3+for+n%3E0)\n\n#### Investigate the convergence or divergence of an infinite sum step by step:\n[Does the sum of 5\\*3^(1 - n) converge?](https:\/\/www.wolframalpha.com\/input?i=Does+the+sum+of+5*3%5E%281+-+n%29+converge%3F)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-proofs)\n\n[Geometry ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-geometry)\n\nExplore two- and three-dimensional shapes and their properties.\n\n#### Compute properties of geometric objects step by step:\n[area of a circle with radius 2](https:\/\/www.wolframalpha.com\/input?i=area+of+a+circle+with+radius+2)\n\n[find the surface area of a sphere with radius = 8](https:\/\/www.wolframalpha.com\/input?i=find+the+surface+area+of+a+sphere+with+radius+%3D+8)\n\n#### Learn how to compute different properties for graphical objects:\n[(2, -4, 7) to (-4, 6, 1) distance](https:\/\/www.wolframalpha.com\/input?i=%282%2C+-4%2C+7%29+to+%28-4%2C+6%2C+1%29+distance)\n\n[express x^2+y^2=3 in polar coordinates](https:\/\/www.wolframalpha.com\/input?i=express+x%5E2%2By%5E2%3D3+in+polar+coordinates)\n\n#### Determine the equation of a line with certain properties using a selected form:\n[equation of line through (2, 5), (4, 1)](https:\/\/www.wolframalpha.com\/input?i=equation+of+line+through+%282%2C+5%29%2C+%284%2C+1%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-geometry)\n\n[Calculus ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-calculus)\n\nGet step-by-step solutions for common calculus problems, like computing limits, derivatives and integrals.\n\n#### Find the derivative using the product rule, chain rule and other methods:\n[compute the derivative of x\\*sin\\[x\\]](https:\/\/www.wolframalpha.com\/input?i=compute+the+derivative+of+x*sin%5Bx%5D)\n\n#### Calculate an indefinite integral by substitution, integration by parts and other methods:\n[integrate sin(x)cos(x)^2](https:\/\/www.wolframalpha.com\/input?i=integrate+sin%28x%29cos%28x%29%5E2)\n\n#### Learn how to take a limit:\n[limit of (x - 3) \/ (x^2 - 2x - 3) as x approaches 3](https:\/\/www.wolframalpha.com\/input?i=limit+of+%28x+-+3%29+%2F+%28x%5E2+-+2x+-+3%29+as+x+approaches+3)\n\n#### Explore step by step a variety of tests to discover local and global extrema:\n[find the extrema of 3x^4 - 4x^3 - 12x^2 + 1](https:\/\/www.wolframalpha.com\/input?i=find+the+extrema+of+3x%5E4+-+4x%5E3+-+12x%5E2+%2B+1)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-calculus)\n\n[Linear Algebra ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-linear-algebra)\n\nSolve problems involving vectors and matrices step by step.\n\n#### Compute properties of a matrix step by step:\n[{{2, 2^2, 8\/4}, {1, - 4, 7 - 4}}](https:\/\/www.wolframalpha.com\/input?i=%7B%7B2%2C+2%5E2%2C+8%2F4%7D%2C+%7B1%2C+-+4%2C+7+-+4%7D%7D)\n\n#### Find the determinant step by step with various methods:\n[determinant of {{1,2,1}, {1,1,0}, {0,1,1}}](https:\/\/www.wolframalpha.com\/input?i=determinant+of+%7B%7B1%2C2%2C1%7D%2C+%7B1%2C1%2C0%7D%2C+%7B0%2C1%2C1%7D%7D)\n\n#### Compute eigenvalues and eigenvectors step by step:\n[eigenvalues {{3,-1},{0,2}}](https:\/\/www.wolframalpha.com\/input?i=eigenvalues+%7B%7B3%2C-1%7D%2C%7B0%2C2%7D%7D)\n\n#### Compute a cross product step by step:\n[(1,2,3)x(3,4,5)](https:\/\/www.wolframalpha.com\/input?i=%281%2C2%2C3%29x%283%2C4%2C5%29)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-linear-algebra)\n\n[Physics ›](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-physics)\n\nSee how to solve physics problems step by step.\n\n#### Do physics calculations step by step:\n[Doppler shift for sound at source frequency 100 kHz for 70 mph](https:\/\/www.wolframalpha.com\/input?i=Doppler+shift+for+sound+at+source+frequency+100+kHz+for+70+mph)\n\n[coulomb's law for 0.1 C and 0.2 C at 2 meters distance](https:\/\/www.wolframalpha.com\/input?i=coulomb%27s+law+for+0.1+C+and+0.2+C+at+2+meters+distance)\n\n[length contraction for a proper length of 1 yard at 0.93c](https:\/\/www.wolframalpha.com\/input?i=length+contraction+for+a+proper+length+of+1+yard+at+0.93c)\n\n[photon energy 435nm](https:\/\/www.wolframalpha.com\/input?i=photon+energy+435nm)\n\n[water flow through a pipe 2\" diameter, 10' length of pipe with 2 kPa pressure difference](https:\/\/www.wolframalpha.com\/input?i=water+flow+through+a+pipe+2%22+diameter%2C+10%27+length+of+pipe+with+2+kPa+pressure+difference)\n\n[More examples](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-physics)\n\n[GoProNow](https:\/\/www.wolframalpha.com\/pro\/pricing\/students)[Learn more about Wolfram\\|Alpha Pro »](https:\/\/www.wolframalpha.com\/pro)","meta_canonical":null}

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Meta Title	Wolfram\|Alpha Examples: Step-by-Step Solutions
Meta Description	Step-by-step solutions for math, chemistry, physics. Calculus, algebra, trigonometry, geometry, number theory, base conversions, statistics. Physics formulas, chemical structures, chemical conversions and reactions. Intercepts, discontinuities, differential equations, linear algebra, mathematical induction.
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Boilerpipe Text	Pro Examples for Solutions for over 200 topics within math, chemistry and physics problems across all levels of education, from elementary school to college. Arithmetic › See how to show your work for arithmetic problems. See step-by-step solutions for doing basic multiplication or division problems: 3 × 5 60431 / 89 See the steps for applying the order of operations for arithmetic: 9 (3+1) + 17 / (6-12) Follow the steps to convert an improper fraction to a mixed number: 87/5 as a mixed number More examples Statistics › See the steps for analyzing data and applying statistical methods. Perform basic statistical analysis on a set of data, seeing the steps along the way: average of {1, 2, 3, 4, 5, 6, 7} mode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20} variance 4, 6, 7, 9, 10, 11 quartile of {1, 2, 3, 4, 5, 6} More examples Differential Equations › Explore step-by-step solutions for solving equations involving derivatives. See how ordinary differential equations are solved: solve y' = y^2 x y'(t) - 2y(t) = 3 e^(2t) t y(t) (1 + t y(t)^2) y'(t) = 1 y''+ y = sin(2x) More examples Proofs › Get step-by-step results for proving mathematical statements. See the steps toward proving a trigonometric identity: (1 + tan(x))/(1 - tan(x)) = (cos(x) + sin(x))/(cos(x) - sin(x)) Prove a sum identity by induction: prove by induction sum of j from 1 to n = n(n+1)/2 for n>0 Prove an inequality by induction: prove by induction (3n)! > 3^n (n!)^3 for n>0 Investigate the convergence or divergence of an infinite sum step by step: Does the sum of 53^(1 - n) converge? More examples Geometry › Explore two- and three-dimensional shapes and their properties. Compute properties of geometric objects step by step: area of a circle with radius 2 find the surface area of a sphere with radius = 8 Learn how to compute different properties for graphical objects: (2, -4, 7) to (-4, 6, 1) distance express x^2+y^2=3 in polar coordinates Determine the equation of a line with certain properties using a selected form: equation of line through (2, 5), (4, 1) More examples Calculus › Get step-by-step solutions for common calculus problems, like computing limits, derivatives and integrals. Find the derivative using the product rule, chain rule and other methods: compute the derivative of xsin[x] Calculate an indefinite integral by substitution, integration by parts and other methods: integrate sin(x)cos(x)^2 Learn how to take a limit: limit of (x - 3) / (x^2 - 2x - 3) as x approaches 3 Explore step by step a variety of tests to discover local and global extrema: find the extrema of 3x^4 - 4x^3 - 12x^2 + 1 More examples Linear Algebra › Solve problems involving vectors and matrices step by step. Compute properties of a matrix step by step: {{2, 2^2, 8/4}, {1, - 4, 7 - 4}} Find the determinant step by step with various methods: determinant of {{1,2,1}, {1,1,0}, {0,1,1}} Compute eigenvalues and eigenvectors step by step: eigenvalues {{3,-1},{0,2}} Compute a cross product step by step: (1,2,3)x(3,4,5) More examples Physics › See how to solve physics problems step by step. Do physics calculations step by step: Doppler shift for sound at source frequency 100 kHz for 70 mph coulomb's law for 0.1 C and 0.2 C at 2 meters distance length contraction for a proper length of 1 yard at 0.93c photon energy 435nm water flow through a pipe 2" diameter, 10' length of pipe with 2 kPa pressure difference More examples Go Pro Now Learn more about Wolfram\|Alpha Pro »
Markdown	![no-js-running-man-logo](https://www.wolframalpha.com/_next/static/images/running-man_d23cee5b24bc562c49a0355e9408ad8c.png)Uh oh! Wolfram\\|Alpha doesn't run without JavaScript. Please enable JavaScript. If you don't know how, you can find instructions[here](https://enable-javascript.com/).Once you've done that, refresh this page to start using Wolfram\\|Alpha. # [All Examples›](https://www.wolframalpha.com/examples)[Pro Features›](https://www.wolframalpha.com/examples/pro-features) Browse Examples Pro Examples for # Step-by-Step Solutions Solutions for over 200 topics within math, chemistry and physics problems across all levels of education, from elementary school to college. ![Step-by-Step Solutions](https://www.wolframcdn.com/examples/proPages/subpageImages/step-by-step-solutions-hero.png) [Arithmetic ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-arithmetic) See how to show your work for arithmetic problems. #### See step-by-step solutions for doing basic multiplication or division problems: [3 × 5](https://www.wolframalpha.com/input?i=3+%C3%97+5) [60431 / 89](https://www.wolframalpha.com/input?i=60431+%2F+89) #### See the steps for applying the order of operations for arithmetic: [9 (3+1) + 17 / (6-12)](https://www.wolframalpha.com/input?i=9+%283%2B1%29+%2B+17+%2F+%286-12%29) #### Follow the steps to convert an improper fraction to a mixed number: [87/5 as a mixed number](https://www.wolframalpha.com/input?i=87%2F5+as+a+mixed+number) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-arithmetic) [Statistics ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-statistics) See the steps for analyzing data and applying statistical methods. #### Perform basic statistical analysis on a set of data, seeing the steps along the way: [average of {1, 2, 3, 4, 5, 6, 7}](https://www.wolframalpha.com/input?i=average+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%2C+7%7D) [mode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20}](https://www.wolframalpha.com/input?i=mode+of+%7B1%2C+5%2C+4%2C+2%2C+3%2C+4%2C+5%2C+11%2C+4%2C+11%2C+20%7D) [variance 4, 6, 7, 9, 10, 11](https://www.wolframalpha.com/input?i=variance+4%2C+6%2C+7%2C+9%2C+10%2C+11) [quartile of {1, 2, 3, 4, 5, 6}](https://www.wolframalpha.com/input?i=quartile+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%7D) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-statistics) [Differential Equations ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-differential-equations) Explore step-by-step solutions for solving equations involving derivatives. #### See how ordinary differential equations are solved: [solve y' = y^2 x](https://www.wolframalpha.com/input?i=solve+y%27+%3D+y%5E2+x) [y'(t) - 2y(t) = 3 e^(2t)](https://www.wolframalpha.com/input?i=y%27%28t%29+-+2y%28t%29+%3D+3+e%5E%282t%29) [t y(t) (1 + t y(t)^2) y'(t) = 1](https://www.wolframalpha.com/input?i=t+y%28t%29+%281+%2B+t+y%28t%29%5E2%29+y%27%28t%29+%3D+1) [y''+ y = sin(2x)](https://www.wolframalpha.com/input?i=y%27%27%2B+y+%3D+sin%282x%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-differential-equations) [Proofs ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-proofs) Get step-by-step results for proving mathematical statements. #### See the steps toward proving a trigonometric identity: [(1 + tan(x))/(1 - tan(x)) = (cos(x) + sin(x))/(cos(x) - sin(x))](https://www.wolframalpha.com/input?i=%281+%2B+tan%28x%29%29%2F%281+-+tan%28x%29%29+%3D+%28cos%28x%29+%2B+sin%28x%29%29%2F%28cos%28x%29+-+sin%28x%29%29) #### Prove a sum identity by induction: [prove by induction sum of j from 1 to n = n(n+1)/2 for n\>0](https://www.wolframalpha.com/input?i=prove+by+induction+sum+of+j+from+1+to+n+%3D+n%28n%2B1%29%2F2+for+n%3E0) #### Prove an inequality by induction: [prove by induction (3n)! \> 3^n (n!)^3 for n\>0](https://www.wolframalpha.com/input?i=prove+by+induction+%283n%29%21+%3E+3%5En+%28n%21%29%5E3+for+n%3E0) #### Investigate the convergence or divergence of an infinite sum step by step: [Does the sum of 5\3^(1 - n) converge?](https://www.wolframalpha.com/input?i=Does+the+sum+of+53%5E%281+-+n%29+converge%3F) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-proofs) [Algebra ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-algebra) Learn how to manipulate and solve expressions with variables step by step. #### Solve equations one step at a time: [find t for t^2 + 3t + 2 = 0](https://www.wolframalpha.com/input?i=find+t+for+t%5E2+%2B+3t+%2B+2+%3D+0) #### Factor polynomials step by step: [factor x^3+2x^2+x+2](https://www.wolframalpha.com/input?i=factor+x%5E3%2B2x%5E2%2Bx%2B2) #### Expand polynomials using the distributive law, the binomial theorem and other procedures: [expanded form of (x + 3)^2](https://www.wolframalpha.com/input?i=expanded+form+of+%28x+%2B+3%29%5E2) #### Learn to rewrite a rational expression using partial fraction decomposition: [partial fraction decomposition 1/(x^2 + 4x + 3)](https://www.wolframalpha.com/input?i=partial+fraction+decomposition+1%2F%28x%5E2+%2B+4x+%2B+3%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-algebra) [Trigonometry ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-trigonometry) Learn the steps for evaluating or manipulating trigonometric expressions. #### Show your work for solving the unknown measurements of a triangle: [right triangle with hypotenuse 2, A = 60 degrees](https://www.wolframalpha.com/input?i=right+triangle+with+hypotenuse+2%2C+A+%3D+60+degrees) #### See the steps for how to find the values of a trigonometric function: [sin(2pi/3)](https://www.wolframalpha.com/input?i=sin%282pi%2F3%29) #### Get step-by-step solutions for applying the sum and difference formulas: [simplify sin(x-y) + sin(x+y)](https://www.wolframalpha.com/input?i=simplify+sin%28x-y%29+%2B+sin%28x%2By%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-trigonometry) [Discrete Mathematics ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-discrete-mathematics) See the steps involved for determining different properties of numbers. #### Find step-by-step solutions for prime factorization, primality testing, GCD and more: [is 73 prime?](https://www.wolframalpha.com/input?i=is+73+prime%3F) [lcm(90, 342)](https://www.wolframalpha.com/input?i=lcm%2890%2C+342%29) [are 12 and 21 coprime?](https://www.wolframalpha.com/input?i=are+12+and+21+coprime%3F) #### Follow the steps to convert between bases: [convert 33ab3653 base 12 to base 10](https://www.wolframalpha.com/input?i=convert+33ab3653+base+12+to+base+10) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-discrete-mathematics) [Chemistry ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-chemistry) Explore a wide array of chemistry topics with step-by-step solutions. #### Get a step-by-step procedure to draw the Lewis structures of molecules: [What is the Lewis structure of NO2](https://www.wolframalpha.com/input?i=What+is+the+Lewis+structure+of+NO2) #### Follow the steps for unit conversion: [convert 5mL benzene to moles](https://www.wolframalpha.com/input?i=convert+5mL+benzene+to+moles) #### Learn how to balance a chemical reaction: [glucose + oxygen -\> water + carbon dioxide](https://www.wolframalpha.com/input?i=glucose+%2B+oxygen+-%3E+water+%2B+carbon+dioxide) #### Compute oxidation states of chemicals: [oxidation states of chromium(III) chloride](https://www.wolframalpha.com/input?i=oxidation+states+of+chromium%28III%29+chloride) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-chemistry) [Geometry ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-geometry) Explore two- and three-dimensional shapes and their properties. #### Compute properties of geometric objects step by step: [area of a circle with radius 2](https://www.wolframalpha.com/input?i=area+of+a+circle+with+radius+2) [find the surface area of a sphere with radius = 8](https://www.wolframalpha.com/input?i=find+the+surface+area+of+a+sphere+with+radius+%3D+8) #### Learn how to compute different properties for graphical objects: [(2, -4, 7) to (-4, 6, 1) distance](https://www.wolframalpha.com/input?i=%282%2C+-4%2C+7%29+to+%28-4%2C+6%2C+1%29+distance) [express x^2+y^2=3 in polar coordinates](https://www.wolframalpha.com/input?i=express+x%5E2%2By%5E2%3D3+in+polar+coordinates) #### Determine the equation of a line with certain properties using a selected form: [equation of line through (2, 5), (4, 1)](https://www.wolframalpha.com/input?i=equation+of+line+through+%282%2C+5%29%2C+%284%2C+1%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-geometry) [Calculus ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-calculus) Get step-by-step solutions for common calculus problems, like computing limits, derivatives and integrals. #### Find the derivative using the product rule, chain rule and other methods: [compute the derivative of x\sin\[x\]](https://www.wolframalpha.com/input?i=compute+the+derivative+of+xsin%5Bx%5D) #### Calculate an indefinite integral by substitution, integration by parts and other methods: [integrate sin(x)cos(x)^2](https://www.wolframalpha.com/input?i=integrate+sin%28x%29cos%28x%29%5E2) #### Learn how to take a limit: [limit of (x - 3) / (x^2 - 2x - 3) as x approaches 3](https://www.wolframalpha.com/input?i=limit+of+%28x+-+3%29+%2F+%28x%5E2+-+2x+-+3%29+as+x+approaches+3) #### Explore step by step a variety of tests to discover local and global extrema: [find the extrema of 3x^4 - 4x^3 - 12x^2 + 1](https://www.wolframalpha.com/input?i=find+the+extrema+of+3x%5E4+-+4x%5E3+-+12x%5E2+%2B+1) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-calculus) [Linear Algebra ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-linear-algebra) Solve problems involving vectors and matrices step by step. #### Compute properties of a matrix step by step: [{{2, 2^2, 8/4}, {1, - 4, 7 - 4}}](https://www.wolframalpha.com/input?i=%7B%7B2%2C+2%5E2%2C+8%2F4%7D%2C+%7B1%2C+-+4%2C+7+-+4%7D%7D) #### Find the determinant step by step with various methods: [determinant of {{1,2,1}, {1,1,0}, {0,1,1}}](https://www.wolframalpha.com/input?i=determinant+of+%7B%7B1%2C2%2C1%7D%2C+%7B1%2C1%2C0%7D%2C+%7B0%2C1%2C1%7D%7D) #### Compute eigenvalues and eigenvectors step by step: [eigenvalues {{3,-1},{0,2}}](https://www.wolframalpha.com/input?i=eigenvalues+%7B%7B3%2C-1%7D%2C%7B0%2C2%7D%7D) #### Compute a cross product step by step: [(1,2,3)x(3,4,5)](https://www.wolframalpha.com/input?i=%281%2C2%2C3%29x%283%2C4%2C5%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-linear-algebra) [Physics ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-physics) See how to solve physics problems step by step. #### Do physics calculations step by step: [Doppler shift for sound at source frequency 100 kHz for 70 mph](https://www.wolframalpha.com/input?i=Doppler+shift+for+sound+at+source+frequency+100+kHz+for+70+mph) [coulomb's law for 0.1 C and 0.2 C at 2 meters distance](https://www.wolframalpha.com/input?i=coulomb%27s+law+for+0.1+C+and+0.2+C+at+2+meters+distance) [length contraction for a proper length of 1 yard at 0.93c](https://www.wolframalpha.com/input?i=length+contraction+for+a+proper+length+of+1+yard+at+0.93c) [photon energy 435nm](https://www.wolframalpha.com/input?i=photon+energy+435nm) [water flow through a pipe 2" diameter, 10' length of pipe with 2 kPa pressure difference](https://www.wolframalpha.com/input?i=water+flow+through+a+pipe+2%22+diameter%2C+10%27+length+of+pipe+with+2+kPa+pressure+difference) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-physics) [GoProNow](https://www.wolframalpha.com/pro/pricing/students)[Learn more about Wolfram\\|Alpha Pro »](https://www.wolframalpha.com/pro) - [Pro](https://www.wolframalpha.com/pro) - [Mobile Apps](https://products.wolframalpha.com/mobile) - [Products](https://products.wolframalpha.com/) - 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Readable Markdown	## Pro Examples for Solutions for over 200 topics within math, chemistry and physics problems across all levels of education, from elementary school to college. ![Step-by-Step Solutions](https://www.wolframcdn.com/examples/proPages/subpageImages/step-by-step-solutions-hero.png) [Arithmetic ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-arithmetic) See how to show your work for arithmetic problems. #### See step-by-step solutions for doing basic multiplication or division problems: [3 × 5](https://www.wolframalpha.com/input?i=3+%C3%97+5) [60431 / 89](https://www.wolframalpha.com/input?i=60431+%2F+89) #### See the steps for applying the order of operations for arithmetic: [9 (3+1) + 17 / (6-12)](https://www.wolframalpha.com/input?i=9+%283%2B1%29+%2B+17+%2F+%286-12%29) #### Follow the steps to convert an improper fraction to a mixed number: [87/5 as a mixed number](https://www.wolframalpha.com/input?i=87%2F5+as+a+mixed+number) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-arithmetic) [Statistics ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-statistics) See the steps for analyzing data and applying statistical methods. #### Perform basic statistical analysis on a set of data, seeing the steps along the way: [average of {1, 2, 3, 4, 5, 6, 7}](https://www.wolframalpha.com/input?i=average+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%2C+7%7D) [mode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20}](https://www.wolframalpha.com/input?i=mode+of+%7B1%2C+5%2C+4%2C+2%2C+3%2C+4%2C+5%2C+11%2C+4%2C+11%2C+20%7D) [variance 4, 6, 7, 9, 10, 11](https://www.wolframalpha.com/input?i=variance+4%2C+6%2C+7%2C+9%2C+10%2C+11) [quartile of {1, 2, 3, 4, 5, 6}](https://www.wolframalpha.com/input?i=quartile+of+%7B1%2C+2%2C+3%2C+4%2C+5%2C+6%7D) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-statistics) [Differential Equations ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-differential-equations) Explore step-by-step solutions for solving equations involving derivatives. #### See how ordinary differential equations are solved: [solve y' = y^2 x](https://www.wolframalpha.com/input?i=solve+y%27+%3D+y%5E2+x) [y'(t) - 2y(t) = 3 e^(2t)](https://www.wolframalpha.com/input?i=y%27%28t%29+-+2y%28t%29+%3D+3+e%5E%282t%29) [t y(t) (1 + t y(t)^2) y'(t) = 1](https://www.wolframalpha.com/input?i=t+y%28t%29+%281+%2B+t+y%28t%29%5E2%29+y%27%28t%29+%3D+1) [y''+ y = sin(2x)](https://www.wolframalpha.com/input?i=y%27%27%2B+y+%3D+sin%282x%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-differential-equations) [Proofs ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-proofs) Get step-by-step results for proving mathematical statements. #### See the steps toward proving a trigonometric identity: [(1 + tan(x))/(1 - tan(x)) = (cos(x) + sin(x))/(cos(x) - sin(x))](https://www.wolframalpha.com/input?i=%281+%2B+tan%28x%29%29%2F%281+-+tan%28x%29%29+%3D+%28cos%28x%29+%2B+sin%28x%29%29%2F%28cos%28x%29+-+sin%28x%29%29) #### Prove a sum identity by induction: [prove by induction sum of j from 1 to n = n(n+1)/2 for n\>0](https://www.wolframalpha.com/input?i=prove+by+induction+sum+of+j+from+1+to+n+%3D+n%28n%2B1%29%2F2+for+n%3E0) #### Prove an inequality by induction: [prove by induction (3n)! \> 3^n (n!)^3 for n\>0](https://www.wolframalpha.com/input?i=prove+by+induction+%283n%29%21+%3E+3%5En+%28n%21%29%5E3+for+n%3E0) #### Investigate the convergence or divergence of an infinite sum step by step: [Does the sum of 5\3^(1 - n) converge?](https://www.wolframalpha.com/input?i=Does+the+sum+of+53%5E%281+-+n%29+converge%3F) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-proofs) [Geometry ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-geometry) Explore two- and three-dimensional shapes and their properties. #### Compute properties of geometric objects step by step: [area of a circle with radius 2](https://www.wolframalpha.com/input?i=area+of+a+circle+with+radius+2) [find the surface area of a sphere with radius = 8](https://www.wolframalpha.com/input?i=find+the+surface+area+of+a+sphere+with+radius+%3D+8) #### Learn how to compute different properties for graphical objects: [(2, -4, 7) to (-4, 6, 1) distance](https://www.wolframalpha.com/input?i=%282%2C+-4%2C+7%29+to+%28-4%2C+6%2C+1%29+distance) [express x^2+y^2=3 in polar coordinates](https://www.wolframalpha.com/input?i=express+x%5E2%2By%5E2%3D3+in+polar+coordinates) #### Determine the equation of a line with certain properties using a selected form: [equation of line through (2, 5), (4, 1)](https://www.wolframalpha.com/input?i=equation+of+line+through+%282%2C+5%29%2C+%284%2C+1%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-geometry) [Calculus ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-calculus) Get step-by-step solutions for common calculus problems, like computing limits, derivatives and integrals. #### Find the derivative using the product rule, chain rule and other methods: [compute the derivative of x\sin\[x\]](https://www.wolframalpha.com/input?i=compute+the+derivative+of+xsin%5Bx%5D) #### Calculate an indefinite integral by substitution, integration by parts and other methods: [integrate sin(x)cos(x)^2](https://www.wolframalpha.com/input?i=integrate+sin%28x%29cos%28x%29%5E2) #### Learn how to take a limit: [limit of (x - 3) / (x^2 - 2x - 3) as x approaches 3](https://www.wolframalpha.com/input?i=limit+of+%28x+-+3%29+%2F+%28x%5E2+-+2x+-+3%29+as+x+approaches+3) #### Explore step by step a variety of tests to discover local and global extrema: [find the extrema of 3x^4 - 4x^3 - 12x^2 + 1](https://www.wolframalpha.com/input?i=find+the+extrema+of+3x%5E4+-+4x%5E3+-+12x%5E2+%2B+1) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-calculus) [Linear Algebra ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-linear-algebra) Solve problems involving vectors and matrices step by step. #### Compute properties of a matrix step by step: [{{2, 2^2, 8/4}, {1, - 4, 7 - 4}}](https://www.wolframalpha.com/input?i=%7B%7B2%2C+2%5E2%2C+8%2F4%7D%2C+%7B1%2C+-+4%2C+7+-+4%7D%7D) #### Find the determinant step by step with various methods: [determinant of {{1,2,1}, {1,1,0}, {0,1,1}}](https://www.wolframalpha.com/input?i=determinant+of+%7B%7B1%2C2%2C1%7D%2C+%7B1%2C1%2C0%7D%2C+%7B0%2C1%2C1%7D%7D) #### Compute eigenvalues and eigenvectors step by step: [eigenvalues {{3,-1},{0,2}}](https://www.wolframalpha.com/input?i=eigenvalues+%7B%7B3%2C-1%7D%2C%7B0%2C2%7D%7D) #### Compute a cross product step by step: [(1,2,3)x(3,4,5)](https://www.wolframalpha.com/input?i=%281%2C2%2C3%29x%283%2C4%2C5%29) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-linear-algebra) [Physics ›](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-physics) See how to solve physics problems step by step. #### Do physics calculations step by step: [Doppler shift for sound at source frequency 100 kHz for 70 mph](https://www.wolframalpha.com/input?i=Doppler+shift+for+sound+at+source+frequency+100+kHz+for+70+mph) [coulomb's law for 0.1 C and 0.2 C at 2 meters distance](https://www.wolframalpha.com/input?i=coulomb%27s+law+for+0.1+C+and+0.2+C+at+2+meters+distance) [length contraction for a proper length of 1 yard at 0.93c](https://www.wolframalpha.com/input?i=length+contraction+for+a+proper+length+of+1+yard+at+0.93c) [photon energy 435nm](https://www.wolframalpha.com/input?i=photon+energy+435nm) [water flow through a pipe 2" diameter, 10' length of pipe with 2 kPa pressure difference](https://www.wolframalpha.com/input?i=water+flow+through+a+pipe+2%22+diameter%2C+10%27+length+of+pipe+with+2+kPa+pressure+difference) [More examples](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-physics) [GoProNow](https://www.wolframalpha.com/pro/pricing/students)[Learn more about Wolfram\\|Alpha Pro »](https://www.wolframalpha.com/pro)
Shard	166 (laksa)
Root Hash	10472823281220336366
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