Pro

Examples for

Step-by-Step Calculus

Derivatives

Find the derivative using the product rule, the 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

See how to evaluate derivatives at a given point:

derivative of x^2+4x-2 at x=3
d/dt arctan(t) at t = 1

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+

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)))

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

Integrals

Calculate an indefinite 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))

See the steps for definite 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

See the steps for multivariate integration over a given domain:

integrate sin(x)cos(y), x = 0 to pi, y = 0 to pi/2
int sin^2 x + y sin z dx dy dz , x=0..pi, y=0..1, z=0..pi

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)

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

Vector Analysis

See the steps to compute the gradient:

Calculate the gradient of (x - 4)*Log[x y]
del (x - 4)*(x - 2)*(y - z)

Learn how to compute the divergence:

div (x^2-y^2, 2xy)
divergence {x+y, x Sin[z], z+y/x^2}

RELATED EXAMPLES

  • Arc Length
  • Area between Curves
  • Continuity
  • Derivatives
  • Inflection Points
  • Integrals
  • Limits
  • Optimization

    • 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

    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

    Sum Convergence

    Learn how to compute the convergence of an infinite sum:

    sum convergence of n
    sum convergence of n^(-2)
    does the sum of 2^(-n) converge
    does the sum of 5*3^(1 - n) converge
    Go Pro NowLearn more about Wolfram|Alpha Pro »