Physics Exercises/Derivative Table

Some standard derivatives used in physics

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

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

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

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

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

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

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

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

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

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

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

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

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

$\frac{d}{d\theta }{e}^{i\theta }=i{e}^{i\theta }$, 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.

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

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

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

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