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