Derivatives of Inverse Trig Functions

Since the trigonometric functions are periodic functions, the function values repeat many times in their respective domains. You can observe the graphs of trigonometric functions fail the horizontal line test for inverse functions. But the inverse of trigonometric functions can be defined by suitably restricting their domains.  The following table lists the inverse trigonometric functions along with their derivatives.

Inverse Trig Functions Table


      Function and its domain                 Derivative and its domain 
 arcsin(x)        -1$\leq$x$\leq$1   $\frac{1}{\sqrt{1-x^{2}}}$             -1 < x < 1 
 arccos(x)         -1$\leq$x$\leq$1  -$\frac{1}{\sqrt{1-x^{2}}}$            -1 < x < 1
 arctan(x)         (-$\infty$,$\infty$)  $\frac{1}{1+x^{2}}$                (-$\infty$,$\infty$) 
 arccsc(x) ) (-$\infty$,-1] U [1,$\infty$)   $\frac{-1}{x\sqrt{x^{2}-1}}$ (-$\infty$,-1] U [1,$\infty$)
 arccsc(x)   (-$\infty$,-1] U [1,$\infty$)  $\frac{-1}{x\sqrt{x^{2}-1}}$ (-$\infty$,-1] U [1,$\infty$)
 arccot(x)       (-$\infty$,$\infty$)  $\frac{-1}{1+x^{2}}$                 (-$\infty$,$\infty$)

                                    Derivatives for composite inverse trigonometric functions 
                                     If u is a differentiable function of x and u' =$\frac{du}{dx}$ 
 $\frac{d}{dx}$ (arcsin u) = $\frac{u'}{\sqrt{1-u^{2}}}$  $\frac{d}{dx}$ (arccos u) = -$\frac{u'}{\sqrt{1-u^{2}}}$
 $\frac{d}{dx}$ (arctan u) = $\frac{u'}{1+u^{2}}$  $\frac{d}{dx}$ (arccot u) = -$\frac{u'}{1+u^{2}}$
 $\frac{d}{dx}$ (arcsec u) = $\frac{u'}{|u|\sqrt{u^{2}-1}}$  $\frac{d}{dx}$ (arccsc u) = $\frac{-u'}{|u|\sqrt{x^{2}-1}}$

Evaluating Inverse Trigonometric Functions


Differentiating Inverse Trigonometric Functions

Inverse trigonometric functions can often be simplified using some trigonometric substitution before finding the derivative.  The simplified function can be differentiated using chain rule to obtain derivative of the given function. The double angle and half angle formulas for trigonometric functions hint at the substitution to be used.

Topics in Derivatives of Inverse Trigonometric Functions