Summary of Derivatives of Inverse Functions
Essential Concepts
- 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
[latex]\frac{d}{dx}(\sin^{-1} x)=\dfrac{1}{\sqrt{1-x^2}}[/latex]
- Derivative of inverse cosine function
[latex]\frac{d}{dx}(\cos^{-1} x)=\dfrac{-1}{\sqrt{1-x^2}}[/latex]
- Derivative of inverse tangent function
[latex]\frac{d}{dx}(\tan^{-1} x)=\dfrac{1}{1+x^2}[/latex]
- Derivative of inverse cotangent function
[latex]\frac{d}{dx}(\cot^{-1} x)=\dfrac{-1}{1+x^2}[/latex]
- Derivative of inverse secant function
[latex]\frac{d}{dx}(\sec^{-1} x)=\dfrac{1}{|x|\sqrt{x^2-1}}[/latex]
- Derivative of inverse cosecant function
[latex]\frac{d}{dx}(\csc^{-1} x)=\dfrac{-1}{|x|\sqrt{x^2-1}}[/latex]
