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.

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

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

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

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