AcademyDifferentiation
Academy
Inverse Derivatives
Level 1 - Math I (Physics) topic page in Differentiation.
Inverse Derivatives
If \(y = f^{-1}(x)\), then \(f(y) = x\).
Differentiating implicitly:
Starting
\[f(y) = x\]
Differentiate
\[f'(y) \cdot \frac{dy}{dx} = 1\]
Result
\[\frac{dy}{dx} = \frac{1}{f'(y)} = \frac{1}{f'(f^{-1}(x))}\]
Example: \(\arcsin(x)\)
If \(y = \arcsin(x)\), then \(\sin(y) = x\)
Example
\[\cos(y) \cdot \frac{dy}{dx} = 1\]
Result
\[\frac{dy}{dx} = \frac{1}{\cos(y)} = \frac{1}{\sqrt{1 - \sin^2(y)}} = \frac{1}{\sqrt{1 - x^2}}\]
General Formula
General
\[\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}\]