Calculus formula sheet
Every derivative, integral, rule and theorem on one page. Tap any name for the full explanation, graph and worked examples, or print it for revision.
Derivatives and integrals of common functions
| Function | Derivative | Integral |
|---|---|---|
| Constant function f(x) = c | 0 | cx + C |
| Linear function f(x) = mx + b | m | \frac{m}{2}x^2 + bx + C |
| Quadratic function f(x) = ax^2 + bx + c, \quad a \ne 0 | 2ax + b | \frac{a}{3}x^3 + \frac{b}{2}x^2 + cx + C |
| Power functions f(x) = x^n | nx^{n-1} | \frac{x^{n+1}}{n+1} + C |
| Polynomial functions f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 | na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + a_1 | \frac{a_n}{n+1}x^{n+1} + \cdots + \frac{a_1}{2}x^2 + a_0x + C |
| Rational functions f(x) = \frac{P(x)}{Q(x)}, \quad Q(x) \ne 0 | \frac{P'(x)Q(x) - P(x)Q'(x)}{Q(x)^2} | \int \frac{A}{x - r}\,dx = A\ln|x - r| + C |
| Reciprocal function f(x) = \frac{1}{x} | -\frac{1}{x^2} | \ln|x| + C |
| Square root function f(x) = \sqrt{x} = x^{1/2} | \frac{1}{2\sqrt{x}} | \frac{2}{3}x^{3/2} + C |
| Absolute value function f(x) = |x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases} | \frac{x}{|x|} | \frac{x|x|}{2} + C |
| Natural exponential function f(x) = e^x | e^x | e^x + C |
| Exponential functions f(x) = b^x, \quad b > 0,\ b \ne 1 | b^x \ln b | \frac{b^x}{\ln b} + C |
| Natural logarithm f(x) = \ln x | \frac{1}{x} | x\ln x - x + C |
| Logarithmic functions f(x) = \log_b x, \quad b > 0,\ b \ne 1 | \frac{1}{x\ln b} | \frac{x\ln x - x}{\ln b} + C |
| Sine function f(x) = \sin x | \cos x | -\cos x + C |
| Cosine function f(x) = \cos x | -\sin x | \sin x + C |
| Tangent function f(x) = \tan x = \frac{\sin x}{\cos x} | \sec^2 x | \ln\lvert\sec x\rvert + C |
| Secant, cosecant and cotangent \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}, \quad \cot x = \frac{\cos x}{\sin x} | \sec x \tan x | \ln\lvert\sec x + \tan x\rvert + C |
| Arcsine f(x) = \arcsin x = \sin^{-1} x | \frac{1}{\sqrt{1 - x^2}} | x\arcsin x + \sqrt{1 - x^2} + C |
| Arccosine f(x) = \arccos x = \cos^{-1} x | -\frac{1}{\sqrt{1 - x^2}} | x\arccos x - \sqrt{1 - x^2} + C |
| Arctangent f(x) = \arctan x = \tan^{-1} x | \frac{1}{1 + x^2} | x\arctan x - \frac{1}{2}\ln(1 + x^2) + C |
| Hyperbolic functions \sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x} | \cosh x | \cosh x + C |
Limit laws
Explanations →Derivative rules
Explanations →
Constant rule
\frac{d}{dx}\, c = 0
Power rule
\frac{d}{dx}\, x^n = n\,x^{n-1}
Constant multiple rule
\frac{d}{dx}\,[c\,f(x)] = c\,f'(x)
Sum and difference rule
\frac{d}{dx}\,[f(x) \pm g(x)] = f'(x) \pm g'(x)
Product rule
\frac{d}{dx}\,[f(x)\,g(x)] = f'(x)\,g(x) + f(x)\,g'(x)
Quotient rule
\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{[g(x)]^2}
Chain rule
\frac{d}{dx}\,f(g(x)) = f'(g(x))\,g'(x)
Implicit differentiation
\frac{d}{dx}\,f(y) = f'(y)\,\frac{dy}{dx}
Logarithmic differentiation
\frac{dy}{dx} = y \cdot \frac{d}{dx}\big[\ln y\big]
Inverse function rule
(f^{-1})'(x) = \frac{1}{f'\big(f^{-1}(x)\big)}
Integration rules
Explanations →
Reverse power rule
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne -1
Linearity of the integral
\int \left[a\,f(x) + b\,g(x)\right]dx = a\int f(x)\,dx + b\int g(x)\,dx
U-substitution
\int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x)
Integration by parts
\int u\,dv = uv - \int v\,du
Definite integral properties
\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx
Series rules
Explanations →
Geometric series formula
\sum_{n=0}^{\infty} a\,r^n = \frac{a}{1 - r}, \quad \lvert r\rvert < 1
nth-term test
\lim_{n\to\infty} a_n \ne 0 \;\Longrightarrow\; \sum a_n \text{ diverges}
Ratio test
L = \lim_{n\to\infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert
Alternating series test
b_n \ge b_{n+1} > 0,\ \lim_{n\to\infty} b_n = 0 \;\Longrightarrow\; \sum (-1)^{n+1} b_n \text{ converges}
Limits and continuity theorems
Explanations →Differential calculus theorems
Explanations →Integral calculus theorems
Explanations →Series and multivariable theorems
Explanations →
Taylor's theorem
f(x) = P_n(x) + \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}
Clairaut's theorem
f_{xy}(a, b) = f_{yx}(a, b)
Green's theorem
\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA
Divergence theorem
\iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \nabla\cdot\mathbf{F}\,dV
Stokes' theorem
\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\nabla\times\mathbf{F})\cdot d\mathbf{S}
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