Math Cheat Sheet

Calculus

a + b = b + a

a \times b = b \times a

(a + b) + c = a + (b + c)

(a \times b) \times c = a \times (b \times c)

(a \times (b + c) = a \times b + a \times c

n! = n (n - 1) (n - 2) … 1

a^m a^n = a^{m + n}

(a b)^m = a^m b^m

(a^m)^n = a^{m \times n}

a^0 = 1

\frac{a^m}{a^n} = a^{m - n}

a^{-m} = \frac{1}{a^m}

x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}

(a + a)^1 = a + b

(a + b)^2 = a^2 + 2 a b + b^2

(a + b)^3 - a^3 + 3 a^2 b + 3 a b^2 + b^3

a^2 - b^2 = (a - b) (a + b)

a^3 - b^3 = (a - b) (a^2 + a b + b^2)

\frac{0}{x} = 0 \text{ if } x \neq 0

a^0 = 1

0^0 = 1

0^x = 0 \text{ if } x \neq 0

a \times 0 = 0

\frac{x}{0} \text{ is undefined}

\sin(\alpha) = \frac{1}{\csc(\alpha)}

\cos(\alpha) = \frac{1}{\sec(\alpha)}

\tan(\alpha) = \frac{1}{\cot(\alpha)}

\cotan(\alpha) = \frac{1}{\tan(\alpha)}

\sin^2(\alpha) + \cos^2(\alpha) = 1

\tan^2(\alpha) + 1 = \sec^2(\alpha)

\cot^2(\alpha) + 1 = \csc^2(\alpha)

\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)

\sin(\alpha - \beta) = \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta)

\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)

\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)

\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \tan(\beta)}

\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha) \tan(\beta)}

\sin \left( \frac{\alpha}{2} \right) = \pm \sqrt{\frac{1 - \cos(\alpha)}{2}}

\cos \left( \frac{\alpha}{2} \right) = \pm \sqrt{\frac{1 + \cos(\alpha)}{2}}

\tan \left( \frac{\alpha}{2} \right) = \frac{1 - \cos(\alpha)}{\sin(\alpha)} = \frac{\sin(\alpha)}{1 + \cos(\alpha)}

\sin(2 \alpha) = 2 \sin(\alpha) \cos(\alpha)

\cos(2 \alpha) = \cos^2(\alpha) - \sin^2(\alpha)

\cos(2 \alpha) = 2 \cos^2(\alpha) - 1

\cos(2 \alpha) = 1 - 2 \sin^2(\alpha)

\tan(2 \alpha) = \frac{2 \tan(\alpha)}{1 - \tan^2(\alpha)}

\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\alpha)} = \frac{c}{\sin(\gamma)}

a^2 = b^2 + c^2 - 2 ⁢b c \cos(\alpha

b^2 = a^2 + c^2 - 2 a c \cos(\beta)

c^2 = a^2 + b^2 - 2 a b \cos(\gamma)

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

\frac{d}{dx}c = 0

\frac{d}{dx}x = 1

\frac{d}{dx}\frac{1}{x} = - \frac{1}{x^2}

\frac{d}{dx}\frac{1}{x^2} = - \frac{2}{x^3}

\frac{d}{dx}\sqrt{x} = \frac{1}{2 \sqrt{x}}

\frac{d}{dx}\frac{1}{\sqrt{x}} = - \frac{1}{2 x \sqrt{x}}

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

\frac{d}{dx}b^x = b^x \ln(b)

\frac{d}{dx}\ln(x) = \frac{1}{x}

\frac{d}{dx}\log_b(x) = \frac{1}{x \ln(b)}

\frac{d}{dx}\sinh(x) = \cosh(x)

\frac{d}{dx}\cosh(x) = \sinh(x)

\frac{d}{dx}\tanh(x) = \operatorname{sech}^2(x)

\frac{d}{dx}\operatorname{arcsinh}(x) = \frac{1}{\sqrt{1 + x^2}}

\frac{d}{dx}\operatorname{arctanh}(x) = \frac{1}{1 - x^2}

\frac{d}{dx}\sin(x) = \cos(x)

\frac{d}{dx}\cos(x) = -\sin(x)

\frac{d}{dx}\tan(x) = \sec^2(x)

\frac{d}{dx}\sec(x) = \tan(x) \sec(x)

\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}

\frac{d}{dx}\arctan(x) = \frac{1}{1 + x^2}

\frac{d}{dx} f(x)^n = n f(x)^{n - 1} f'(x)

\frac{d}{dx} \frac{1}{f(x)} = \frac{-g'(x)}{g(x)^2}

\frac{d}{dx} \ln | f(x) | = \frac{f'(x)}{f(x)}

\frac{d}{dx} e^{f(x)} = f'(x) e^{f(x)}

\int x^n dx = \frac{1}{n + 1} x^{n + 1} + C

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

\int c dx = c x + C

\int x dx = \frac{1}{2} x^2 + C

\int x^2 dx = \frac{1}{3} x^3 + C

\int \frac{1}{x^2} dx = - \frac{1}{x} + C

\int \sqrt{x} dx = \frac{2}{3} x \sqrt{x} + C

\int \frac{1}{\sqrt{x}} dx = 2 \sqrt{x} + C

\int \frac{1}{1 + x^2} dx = \arctan(x) + C

\int \frac{1}{\sqrt{1 - x^2}} dx = \arcsin(x) + C

\int \ln(x) dx = x \ln(x) - x + C

\int x^n \ln(x) dx = \frac{x^{n + 1}}{n + 1} \ln(x) - \frac{x^{n + 1}}{(n + 1)^2} + C

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

\int b^x dx = \frac{1}{\ln(b)} b^x

\int \sinh(x) dx = \cosh(x) + C

\int \cosh(x) dx = \sinh(x) + C

\int \sin(x) dx = -\cos(x) + C

\int \cos(x) dx = \sin(x) + C

\int \tan(x) dx = \ln(|\sec(x)|) + C

\int \sec(x) dx = \ln(|\tan(x) + \sec(x)|) + C

\int \sin^2(x) dx = \frac{1}{2} \left( x - \sin(x) \cos(x) \right) + C

\int \cos^2(x) dx = \frac{1}{2} \left( x + \sin(x) \cos(x) \right) + C

\int \tan^2(x) dx = \tan(x) - x + C

\int \sec^2(x) dx = \tan(x) + C