Derivative

Definition

\[\begin{split}f'(x) &= \lim_{h\to0} \frac{f(x+h)-f(x)}{h} \\ \frac d {dx}(c) &=0 \\ \frac d {dx} (x) &=1 \\\end{split}\]

Derivative Rule

\[\begin{split}(c f)' &= c f'(x) \\ (f \pm g)' &= f'(x) \pm g'(x) \\ (f g)' &= f'g + fg' &\text{- Product Rule}\\ (\frac f g)' &=\frac{f'g-fg'} {g^2} &\text{- Quotient Rule}\\ \frac d {dx}(x^n) &=nx^{n-1} &\text{- Power Rule}\\ \frac {d}{dx}(f(g(x))) &= f'(g(x))g'(x) &\text{- Chain Rule}\\ \frac {dx}{dy} &= \frac{dx}{du} \frac{du}{dy} &\text{- Chain Rule}\\\end{split}\]

Common

\[\begin{split}\frac{d}{dx} (ln(x)) &= \frac 1 x\\ \frac{d}{dx} (e^x) &= e^x \\ \frac{d}{dx} (log_b(x)) &= \frac 1 {x ln(b)}\end{split}\]

Tirgonometrics Derivatives

\[\begin{split}csc x = \frac 1 {sin x}\\ sec x = \frac 1 {cos x}\\ cot x = \frac 1 {tan x}\\\end{split}\]

\[\begin{split}\frac {d}{dx} (sin x) &=cos x \\ \frac {d}{dx} (cos x) &=-sinx \\ \frac {d}{dx} (tan x) &={sec}^2 x \\ \frac {d}{dx} (csc x) &= - csc x cot(x) \\ \frac {d}{dx} (cse x) &= cse x tan x \\ \frac {d}{dx} (cot x) &= {csc}^2 x\end{split}\]