Wolfram|Alpha Examples: Step-by-Step Calculus

Derivatives

Find the derivative using the product rule, chain rule and other methods:

d/dx sqrt(x)

derivative of x^4 + 9x^3 + 7x - 2

slope of log(t) / (t^2 + 1)

Find partial derivatives of multivariate expressions:

partial derivative of exp(-x^2 - y^2)

partial derivative of cos( 3 pi /4 * x + 5 pi /3 * y + pi / 2 * z) with respect to y

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Indefinite Integrals

Calculate an integral by substitution, integration by parts and other methods:

integrate sin(x)cos(x)^2

integral of sqrt(a^2 - x^2)

show the integration: arcsec(sqrt(t))

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Definite Integrals

See the steps for integration over an interval:

integrate sin(t)^2cos(t) from 0 to pi/2

integrate 1/(x^2 + 1), x = -infinity ... infinity

integrate 1/(cos(x) + 2) from 0 to 2pi

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Limits

Learn how to take a limit:

limit of (x - 3) / (x^2 - 2x - 3) as x approaches 3

take the limit as x goes to infinity: (1 + 1/x)^x

limit t*tan(t) as t->pi/2 from the left

lim (x^2 + 2x + 3)/(x^2 - 2x - 3) as x->3+

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Optimization

Find local and global extrema using a variety of tests:

local extrema of x^3 - 3x + 1

find the extrema of 3x^4 - 4x^3 - 12x^2 + 1

maximize sin(x)

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Tangent Lines & Planes

Find a tangent line to a curve using the derivative at a point:

tangent line to sin(x) - 1 at x=0

tangent line of y=x^2 at x=1

Calculate a multivariate tangent plane by examining partial derivatives:

tangent plane to z=x^2+y^2 at x=1,y=1

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Discontinuities

Get hints for discontinuities:

where is (x^2 - 4)/(x^2 - 2x - 3) discontinuous?

discontinuities of sin((x^2-x)/(x-1)) + log(sqrt(abs(x)))

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Inflection Points

Calculate inflection points step by step:

inflection points x^3 - 3 x^2 + 1

points of inflection t * cbrt(t^2 - 4)

inflection points of 3sin(2x)

inflection pts sgn(x)|x|^(3/2) + x^3 - 3x^2 + 1 over -2 < x < 3

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Area between Curves

Learn how to compute the area between curves:

area between 1/2x(x-2)(x+2) and x

area of the region between y=x and y=x^2 for -1/2<x<1

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Arc Length

Calculate the arc length of a curve step by step:

arc length of Sin[x] for x between 0 and 2pi

Specify the curve in polar coordinates:

arc length of r=theta^2 for 0<theta<2pi

Specify the curve parametrically:

arc length of {Sin[x], Cos[x], x, x} for 0<x<3pi

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