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



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

Quantitative Aptitude
Reasoning
Programming
Interview