Calculus basics formulas

The first calculus formulas: the rules for differentiating powers, products, quotients, compositions, trig, exponential and log functions; tangent lines and the difference quotient that defines the derivative; and integrals for areas and averages. Each calculator works at a point you choose and shows the rule being applied, so you can check homework line by line. Angles are in radians throughout.

Printable cheat sheet Formula finder

Power rule for derivatives

d/(dx)(axⁿ) = nax^(n − 1)

To differentiate a power of x, bring the power down as a multiplier and lower it by one.

Product rule

(uv)′ = u′v + uv′

Differentiates a product of two functions: each one's derivative times the other, added.

Quotient rule

(u/v)′ = (u′v − uv′)/(v²)

Differentiates one function divided by another.

Chain rule

dy/(dx) = dy/(du)·du/(dx)

Differentiates a function of a function: outside derivative times inside derivative.

Derivatives of sin and cos

d/(dx)sin (kx) = kcos (kx), d/(dx)cos (kx) = −ksin (kx)

Sine differentiates to cosine and cosine to minus sine, with x in radians.

Derivatives of eˣ and ln x

d/(dx)e^(kx) = ke^(kx), d/(dx)ln x = 1/x

$e^x$ is its own derivative, and the derivative of $\ln x$ is $1/x$ .

Power rule for integrals

∫xⁿ dx = (x^(n + 1))/(n + 1) + C, n ≠ − 1

To integrate a power of x, raise the power by one and divide by the new power.

Integral of eᵏˣ

∫e^(kx) dx = 1/ke^(kx) + C

The integral of an exponential is the same exponential divided by its rate k.

Definite integral (area under a curve)

∫ₐ^bf(x) dx = F(b) − F(a)

The signed area between a curve and the x-axis from a to b, from any antiderivative F.

Average value of a function

f̄ = 1/(b − a)∫ₐ^bf(x) dx

The height of the rectangle on [a, b] with the same area as the region under the curve.

Equation of a tangent line

y = f(x₀) + f′(x₀)(x − x₀)

The straight line that touches a curve at one point with the same slope as the curve there.

Difference quotient

f′(x) = lim[h → 0] (f(x + h) − f(x))/h

The slope of a secant line over a small step h; its limit as h → 0 is the derivative.