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Example Solutions for

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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