Summary of Derivatives of Inverse Functions

The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.
We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

Key Equations

Derivative of inverse sine function
$\frac{d}{dx}(\sin^{-1} x)=\dfrac{1}{\sqrt{1-x^2}}$
Derivative of inverse cosine function
$\frac{d}{dx}(\cos^{-1} x)=\dfrac{-1}{\sqrt{1-x^2}}$
Derivative of inverse tangent function
$\frac{d}{dx}(\tan^{-1} x)=\dfrac{1}{1+x^2}$
Derivative of inverse cotangent function
$\frac{d}{dx}(\cot^{-1} x)=\dfrac{-1}{1+x^2}$
Derivative of inverse secant function
$\frac{d}{dx}(\sec^{-1} x)=\dfrac{1}{|x|\sqrt{x^2-1}}$
Derivative of inverse cosecant function
$\frac{d}{dx}(\csc^{-1} x)=\dfrac{-1}{|x|\sqrt{x^2-1}}$