Calculus Formula Sheet

A quick-reference sheet of essential calculus formulas. Covers limits, derivative rules, integral formulas, the Fundamental Theorem, and series convergence tests. Each formula links to its full definition page.

Limits

Limit Definition of Derivative

f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

L'Hôpital's Rule

\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}

Special Limit

\lim_{x\to 0}\frac{\sin x}{x} = 1

Derivative Rules

Power Rule

\frac{d}{dx}[x^n] = nx^{n-1}

Product Rule

(fg)' = f'g + fg'

Quotient Rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)

Exponential

\frac{d}{dx}[e^x] = e^x

Natural Log

\frac{d}{dx}[\ln x] = \frac{1}{x}

Trig Derivatives

sin x

\frac{d}{dx}[\sin x] = \cos x

cos x

\frac{d}{dx}[\cos x] = -\sin x

tan x

\frac{d}{dx}[\tan x] = \sec^2 x

sec x

\frac{d}{dx}[\sec x] = \sec x\tan x

Integral Formulas

Power Rule

\int x^n\,dx = \frac{x^{n+1}}{n+1}+C \quad(n\neq -1)

Exponential

\int e^x\,dx = e^x + C

1/x

\int \frac{1}{x}\,dx = \ln|x| + C

sin x

\int \sin x\,dx = -\cos x + C

cos x

\int \cos x\,dx = \sin x + C

sec² x

\int \sec^2 x\,dx = \tan x + C

Fundamental Theorem of Calculus

Part 1

\frac{d}{dx}\int_a^x f(t)\,dt = f(x)

Part 2

\int_a^b f(x)\,dx = F(b) - F(a)

Integration Techniques

u-Substitution

\int f(g(x))g'(x)\,dx = \int f(u)\,du

Integration by Parts

\int u\,dv = uv - \int v\,du

Series & Convergence

Taylor Series

f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n

Geometric Series

\sum_{n=0}^{\infty}ar^n = \frac{a}{1-r},\;|r|<1

p-Series

\sum_{n=1}^{\infty}\frac{1}{n^p}\text{ converges iff }p>1