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Examples for

Step-by-Step Calculus

Derivatives

Explore the steps for derivative calculations using the power rule:

derivative of 1/x^2

Find the derivative using the product rule:

compute the derivative of x*sin[x]

f'(x) if f(x) = (x+1)(x^2 + 3)

Compute derivatives using the quotient rule:

d/dx of (x^2-x)/(x^2+3)

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

See step by step how to apply the chain rule:

derivative of exp(sin(x))

derivative of (cos(log(x)))^3

Use the limit definition to compute derivatives:

find d/dx for (x^2)/x^3

Find partial derivatives of multivariate expressions:

partial derivatives 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

Perform logarithmic differentiation to compute a derivative:

first derivative of 1/x^3

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))

Use u-substitution to compute an integral step by step:

integrate x*cos(x^2+1)

integrate (2x^3+1)^7(x^2) for 0<x<1

Compute an integral by parts:

integrate by parts x*cos(x)

Integrate using trigonometric substitution:

integrate tan[x]

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

integrate x^2 sin y dx dy, x=0..y, y=0..pi

Tangent Lines & Planes

Find a line tangent 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

area under 1/x for 1<x<5

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}

Calculate the curl:

curl [3yz, 2xz, 5yx]

curl of {y^2 cos z, z sin y, e^x}

Applications of Derivatives

Find the line tangent to a curve at a point step by step:

tangent line to x^2-1 at x=5

compute the line tangent to (x^3-1)(x+2) at x=1

Compute the local maxima or minima of a function:

local minimum for y==x^2-x+1

local maximum x^5 - 10x^3 + 30x

Show the steps for finding the global minima or maxima of a function:

global minima for y==x^4-2x^2+1

maximize x^5-2x^2+1

Discover how to calculate the extrema of a function on a given interval:

maximize x^5-2x^2+1 on -1<x<1

minimize -2Cos[x] on -1<x<1

Find the inflection points of a function:

what are the inflection points of sin(x)

x^3-3x^2-4x+1 inflection points

Limits

Learn how to take a limit:

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

Calculate a limit approaching infinity:

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

Perform a one-sided limit step by step:

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

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

Use L'Hôpital's rule to compute a limit:

Limit of 5 Log[x]/(x - 1) as x->1

See the steps to use the squeeze theorem on a limit:

limit of sin[x]/x as x->0

Continuity

Get step-by-step solutions for finding discontinuities:

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

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

Determine the continuity of a function at a point:

(x-5)/(x-5) continuous for x = 5

is 3 Cos[5 x] continuous at x = 1?

is Piecewise[{{x^2, x < 0},{4x + 8, x >= 0}}] continuous at x = 0?

Arc Length

Calculate the arc length of a curve step by step:

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

Specify a curve in polar coordinates:

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

Specify a curve parametrically:

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

Sum Convergence

Learn how to compute the convergence of an infinite sum:

sum convergence of n

does the sum of 2^(-n) converge

does the sum of 5*3^(1 - n) converge

Determine whether a series converges step by step:

does the sum of 1/n^2 converge?

convergence of the sum (2/3)^x

Test whether a series converges absolutely:

does the sum of (-2)^(-n) converge absolutely

Explore divergent series:

sum convergence of 1/n

sum convergence of u/(u^2+1) for u=1 to infinity

RELATED EXAMPLES

Area between Curves

Inflection Points

Optimization

Explore step by step a variety of tests to discover local and global extrema:

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

Applications of Integration

Find the average value of an equation on an interval:

average value of 1/x for 1<x<5

Look through the steps to find the average value of a derivative on an interval:

average value of the derivative of Sin(x) for x=1 to 5

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