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# Calculus Basics

## Basic idea

Differentiation measures instantaneous rate of change; integration sums infinitesimal contributions. The two are inverse operations.

## Key formulas

* Limit definition of derivative: $f'(x)=\lim\_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$
* Power rule: $\dfrac{d}{dx}x^n = n x^{n-1}$
* Product rule: $(fg)' = f'g + fg'$
* Quotient rule: $(f/g)' = (f'g - fg')/g^2$
* Chain rule: $(f\circ g)'(x)=f'(g(x)),g'(x)$
* Fundamental theorem of calculus: $\int\_a^b f'(x),dx = f(b)-f(a)$
* $\int x^n,dx = \dfrac{x^{n+1}}{n+1} + C;(n\ne -1)$, $\int \frac{1}{x},dx=\ln|x|+C$
* $\dfrac{d}{dx}e^x=e^x$, $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$
* $\dfrac{d}{dx}\sin x=\cos x$, $\dfrac{d}{dx}\cos x=-\sin x$
* Taylor series: $f(x)=\sum\_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$