Physics Exercises/Derivative Table

Some standard derivatives used in physics

${\displaystyle {\frac {d}{d\theta }}\tan \theta =\sec ^{2}\theta }$

${\displaystyle {\frac {d}{d\theta }}\cot \theta =-\csc ^{2}\theta }$

${\displaystyle {\frac {d}{d\theta }}\sec \theta =\sec \theta \tan \theta }$

${\displaystyle {\frac {d}{d\theta }}\csc \theta =-\csc \theta \cot \theta }$

${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}$

${\displaystyle {\frac {d}{d\theta }}\sin k\theta =k\cos k\theta }$

${\displaystyle {\frac {d}{d\theta }}\cos k\theta =-k\sin k\theta }$

${\displaystyle {\frac {d}{d\theta }}{\sqrt {\theta }}={\frac {1}{2{\sqrt {\theta }}}}}$

${\displaystyle {\frac {d}{dx}}e^{kx}=ke^{kx}}$

${\displaystyle {\frac {d}{d\theta }}\ln \theta ={\frac {1}{\theta }}}$

${\displaystyle {\frac {d}{d\theta }}\sin ^{-1}\theta ={\frac {1}{\sqrt {1-\theta ^{2}}}}}$

${\displaystyle {\frac {d}{d\theta }}\cos ^{-1}\theta =-{\frac {1}{\sqrt {1-\theta ^{2}}}}}$

${\displaystyle {\frac {d}{d\theta }}\tan ^{-1}\theta ={\frac {1}{1+\theta ^{2}}}}$

${\displaystyle {\frac {d}{d\theta }}e^{i\theta }=ie^{i\theta }}$, where ${\displaystyle i={\sqrt {-1}}}$

The following deal with the variable, ${\displaystyle \theta }$, and a function, ${\displaystyle \delta }$, of ${\displaystyle \theta }$, and are examples of the chain rule in action.

${\displaystyle {\frac {d}{d\theta }}\sin \delta =(\cos \delta ){\frac {d}{d\theta }}\delta }$

${\displaystyle {\frac {d}{d\theta }}\delta ^{4}=(4\delta ^{3}){\frac {d}{d\theta }}\delta }$

${\displaystyle {\frac {d}{d\theta }}\theta ^{3}\delta =3\theta ^{2}\delta +(\theta ^{3}){\frac {d}{d\theta }}\delta }$

${\displaystyle {\frac {d}{d\theta }}\theta ^{2}\delta ^{2}=2\theta \delta ^{2}+(2\theta ^{2}\delta ){\frac {d}{d\theta }}\delta }$