# Derivatives

$Df(x)=f'(x)=\frac{df}{dx}=\frac{d}{dx}f=\dot{f}(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$

Introduced in order to fix the idea of velocity: $v = \lim_{\Delta t \to 0}\frac{\Delta s}{\Delta t}=\frac{ds}{dt}$

Working example:

$s(t) = At^3 + Bt + C$

$s + \Delta s = A ( t + \Delta t )^3 + B ( t + \Delta t ) + C =$

$= A t^3 + 3A t^2\Delta t + 3A t ( \Delta t) ^2 + A ( \Delta t ) ^3 + Bt + B\Delta t + C =$

$= ( A t^3 + Bt + C ) +3 A t^2\Delta t + 3A t ( \Delta t) ^2 + A ( \Delta t ) ^3 + B\Delta t =$

$= s(t) + 3A t^2\Delta t + 3A t ( \Delta t) ^2 + A ( \Delta t ) ^3 + B\Delta t \rightarrow$

$\Delta s = 3A t^2\Delta t + 3A t ( \Delta t) ^2 + A ( \Delta t ) ^3 + B\Delta t$

$\frac{\Delta s}{\Delta t} = 3A t^2 + 3A t \Delta t+ A ( \Delta t ) ^2 + B$

$\lim_{\Delta t \to 0}\frac{\Delta s}{\Delta t}= 3A t^2 + B$

Which is exactly $df/dt$.

$D \sqrt{x^2-1}$

$D \sin{(x^2-1)}$

$D \tan\sqrt{x}$

$D \ln|\cos{x}|$ and $D \ln|\sin{x}|$

$D \tan e^{\sqrt{x}}$

And for the die-hard:

$D e^{\imath x}\sin x$

You can check them using mathematica or sage (sage is free).