common derivatives

Exponential

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

$a^x$

$\frac{d}{dx}{a}^x=a^x\cdot \ln a$

Trigonometric Functions

$\frac{d}{dx}\sin x=\cos x$

$\frac{d}{dx}\cos x=-\sin x$

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

$\frac{d}{dx}\sec x=\sec x\tan x$

$\frac{d}{dx}\csc x=-\csc x\cot x$

$\frac{d}{dx}\cot x=-{\csc }^{2}x$

Inverse Trigonometric Functions

Note: there are bounds

$\frac{d}{dx}{\sin }^{-1}u=\frac{1}{\sqrt{1-u^2}}\cdot \frac{du}{dx}$

$\frac{d}{dx}{\cos }^{-1}u=-\frac{1}{\sqrt{1-u^2}}\cdot \frac{du}{dx}$

$\frac{d}{dx}{\sec }^{-1}u=\frac{1}{|u|\sqrt{u^2-1}}\frac{du}{dx}$

$\frac{d}{dx}{\csc }^{-1}u=-\frac{1}{|u|\sqrt{u^2-1}}\frac{du}{dx}$

$\frac{d}{dx}{\tan }^{-1}u=\frac{1}{1+u^2}\cdot \frac{du}{dx}$

$\frac{d}{dx}{\cot }^{-1}u=-\frac{1}{1+u^2}\cdot \frac{du}{dx}$

Logarithm

$(\ln f{)}^{\prime }=\frac{f^{\prime }}{f}$

$({\log }_{b}f{)}^{\prime }=\frac{f^{\prime }}{x\ln b}$

Use the change-of-base formula for other bases.