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Calculus Cheat Sheet

A quick reference for BC calculus

Created 10/4/2021

Derivatives

General Form

$$\displaystyle f'(x) = \lim\limits_{h \to \infty} \frac{f(x+h)- f(x)}{h}$$

General

$$f(x)$$ $$f'(x)$$
$$x^n$$ $$nx^{n-1}$$
$$f(g(x))$$ $$f'(g(x))\cdot g'(x)$$
$$\frac{f(x)}{g(x)}$$ $$\frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}$$
$$f(x)\cdot g(x)$$ $$f'(x)g(x)+f(x)g'(x)$$

Logarithmic

$$f(x)$$ $$f'(x)$$
$$a^x$$ $$ln(a)\cdot a^x$$
$$\log_{a} x$$ $$\frac{1}{x\cdot ln(a)} $$

Exponential

$$f(x)$$ $$f'(x)$$
$$ln(x)$$ $$1/x$$
$$e^x$$ $$e^x$$

Trig

$$f(x)$$ $$f'(x)$$
$$\sin(x)$$ $$\cos(x)$$
$$\cos(x)$$ $$-\sin(x)$$
$$\tan(x)$$ $$\sec^2(x)$$
$$\csc(x)$$ $$-\csc(x)\cot(x)$$
$$\sec(x)$$ $$\sec(x)\tan(x)$$
$$\cot(x)$$ $$-\csc(x)\sec(x)$$

Inverse

$$f'(x) = \frac{1}{f^{-1}{'}(f(x))}$$

Inverse Trig

$$f(x)$$ $$f'(x)$$
$$\sin^{-1}(x)$$ $$\frac{1}{\sqrt{1-x^2}}$$
$$\cos^{-1}(x)$$ $$-\frac{1}{\sqrt{1-x^2}}$$
$$\tan^{-1}(x)$$ $$\frac{1}{1+x^2}$$
$$\csc^{-1}(x)$$ $$-\frac{1}{|x|\sqrt{x^2-1}}$$
$$\sec^{-1}(x)$$ $$\frac{1}{\sqrt{x^2(x^2-1)}}$$
$$cot^{-1}(x)$$ $$-\frac{1}{x^2+1}$$

Integrals

General Form

$$\displaystyle\int_a^b f(x)dx = \lim\limits_{n \to \infty}\sum_{n=1} ^{\infty} (\frac{b-a}{n})f(\frac{b-a}{n}k+a)$$