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# Math Formulas: Common Derivatives

### Basic Properties of Derivatives

 $$\left(c \cdot f(x)\right)' = c \cdot f'(x)$$
 $$\left(f \pm g \right)' = f' \pm g'$$

Product rule

 $$(f \cdot g)' = f' \cdot g + f \cdot g'$$

Quotient rule

 $$\left( \frac{f}{g} \right)' = \frac{ f'\cdot g - f \cdot g' }{g^2}$$

Chain rule

 $$\left( f \left(g(x) \right) \right)' = f'(g(x)) \cdot g'(x)$$

### Common Derivatives

 $$\frac{d}{dx} (C) = 0$$
 $$\frac{d}{dx} (x) = 0$$
 $$\frac{d}{dx} (x^n) = n \cdot x^{n-1}$$
 $$\frac{d}{dx} (\sin x) = \cos x$$
 $$\frac{d}{dx} (\cos x) = -\sin x$$
 $$\frac{d}{dx} (\tan x) = \frac{1}{\cos^2x}$$
 $$\frac{d}{dx} ( \sec x) = \sec x \cdot \tan x$$
 $$\frac{d}{dx} (\csc x) = - \csc x \cdot \cot x$$
 $$\frac{d}{dx} (\cot x) = -\frac{1}{ \sin^2x }$$
 $$\frac{d}{dx} (\arcsin x) = \frac{1}{ \sqrt{1-x^2} }$$
 $$\frac{d}{dx} (\arccos x) = -\frac{1}{\sqrt{1-x^2}}$$
 $$\frac{d}{dx} (\arctan x) = \frac{1}{1+x^2}$$
 $$\frac{d}{dx} (a^x) = a^x \cdot \ln a$$
 $$\frac{d}{dx} (e^x) = e^x$$
 $$\frac{d}{dx} (\ln x) = \frac{1}{x} , x > 0$$
 $$\frac{d}{dx} (\ln |x|) = \frac{1}{x} , x \ne 0$$
 $$\frac{d}{dx} \left( \log_a x \right) = \frac{1}{x\cdot \ln a} , x > 0$$