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Wize AP Calculus (BC) Textbook > Differentiation 2: Composite, Implicit, & Inverse Functions

Derivatives of Inverse Functions

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Derivatives of Inverse Functions

Letf(x)f\left(x\right) be a differentiable function. If f′(f−1(a))≠0f'(f^{-1}(a))\neq 0 and is defined, then f−1(x)f^{-1}\left(x\right)is differentiable at x=ax=a, and
(f−1)′(a)=1f′(f−1(a))\boxed{(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}}

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Example: Derivatives of Inverses

Let f(x)=x4−x+3f(x)=x^4-x+3. Find (f−1)′(17).(f^{-1})'(17). Hint f−1(17)=2f^{-1}(17)=2.


(f−1)′(17)=1f′(f−1(17))=1f′(2)f′(x)=4x3−1f′(2)=4(2)3−1=31(f−1)′(17)=131\begin{aligned} &(f^{-1})'(17)=\frac{1}{f'(f^{-1}(17))}\\&=\frac{1}{f'(2)}\\ \\ &f'(x)=4x^3-1\\ &f'(2)=4(2)^3-1=31\\ &(f^{-1})'(17)=\frac{1}{31} \end{aligned}


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Determine the derivative of arccos⁡x\arccos{x} using the differentiation rule for inverse functions:
ddx[f−1(x)]=1f′(f−1(x))\mydd{f^{-1}(x)}{x}[bt] = \frac{1}{f'(f^{-1}(x ))}