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]
Candela Citations
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- Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/details/books/calculus-volume-1. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction