Physics Exercises/Derivative Table

Some standard derivatives used in physics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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