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}

See: Zeno’s paradoxes

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.

 

Enjoy your derivatives

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).

Useful Links

Mathematica for students http://www.wolfram.com/mathematica/how-to-buy/education/students.html

Wolfram:Alpha http://www.wolframalpha.com

Khan Academy http://www.khanacademy.org

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One thought on “Derivatives

  1. Pingback: Even and Odd Functions « Beyond the Standard Model Pub

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