Differentiation

• First derivatives

  $f'\left(x\right)<0$ => f is decreasing

  $f'\left(x\right)=$$0$ => f(x) is a stationary point

  $f'\left(x\right)>0$ => f is increasing
• Second derivatives

  $f''\left(x\right)<0$ => f is concave, has a max

  $f''\left(x\right)=$$0$ => point of inflection

  $f''\left(x\right)>0$ => f is convex, has a min
• Functions and their derivatives

  $e^x$ => $e^x$
• Functions and their derivatives

  $\ln x$ => $\frac{1}{x}$
• Functions and their derivatives

  $\sin x$ => $\cos x$
• Functions and their derivatives

  $\cos x$ => $-\sin x$
• Functions and their derivatives

  $\tan x$ => $\sec^2x$
• Functions and their derivatives

  $\sec x$ => $\sec x\tan x$
• Functions and their derivatives

  $\cot x$ => $-\operatorname{cosec}^2x$
• Functions and their derivatives

  $\operatorname{cosec}x$ => $-\operatorname{cosec}x\cot x$
• Product rule

  $uv$ => $uv'\ +$$\ u'v$
• Quotient rule

  $\frac{u}{v}$ => $\frac{vu'-uv'}{v^2}$
• Parametric

  $\frac{dy}{dx}=$ $\frac{dy}{dt}\div\frac{dx}{dt}$
• Connected rates of change

  $\frac{dv}{dt}=$ $\frac{dv}{?}\cdot\frac{?}{dt}$
• Rate means something over dt i.e. $\frac{?}{dt}$