Step By Step Calculus » 10.5 - Derivatives for Trigonometric and Inverse Functions

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Synopsis
The following table lists the derivatives of the 6 principal trigonometric functions and their inverses.
\newcommand{\T}{\rule{0pt}{3.1ex}} \newcommand{\B}{\rule[-2.0ex]{0pt}{0pt}} \begin{tabular}{|l|c|c||l|c|c|} \hline Function & Derivative &$\begin{array}{c}\textrm{Derivative}\\\textrm{domain}\end{array}$ & Function & Derivative &$\begin{array}{c}\textrm{Derivative}\\\textrm{domain}\end{array}$\\ \hline \T\B$\sin x$ & $\cos x$ & $\mathbb{R}$ & $\cos x$ & $-\sin x$ & $\mathbb{R}$\\ \hline \T\B$\tan x$ & $\sec^{2}x$ & $x\neq n\pi+\frac{\pi}{2}, n\in\mathbb{Z}$& $\cot x$ & $-\csc^{2}x$ & $x\neq n\pi, n\in\mathbb{Z}$\\ \hline \T\B$\sec x$ & $\sec x\tan x$ & $x\neq n\pi+\frac{\pi}{2}, n\in\mathbb{Z}$ & $\csc x$ & $-\csc x\cot x$& $x\neq n\pi, n\in\mathbb{Z}$ \\ \hline \T\B$\sin^{-1}x$ & $\dfrac{1}{\sqrt{1-x^{2}}}$ & $(-1,1)$& $\cos^{-1}x$ & $-\dfrac{1}{\sqrt{1-x^{2}}}$ & $(-1,1)$\\ \hline \T\B$\tan^{-1}x$ & $\dfrac{1}{1+x^{2}}$ & $\mathbb{R}$& $\cot^{-1}x$ & $-\dfrac{1}{1+x^{2}}$ & $\mathbb{R}$\\ \hline \T\B$\sec^{-1}x$ & $\dfrac{1}{|x|\sqrt{x^{2}-1}}$ & $[-1,1]^C$& $\csc^{-1}x$ & $-\dfrac{1}{|x|\sqrt{x^{2}-1}}$ & $[-1,1]^C$\\ \hline \end{tabular}