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Wize University Calculus 1 Textbook > Derivatives

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 ))}

Extra Practice
Practice: Tangent to the Inverse
Q:\textbf{Q:} Let f(x)=x3f(x)=x^3. Find the slope of the tangent line to f−1 at x=8=f(2)f^{-1}\text{ at }x=8=f(2).
Derivatives of Inverse Functions
Suppose f(x)=x3−3x2−x+4,x≥3f(x)=x^3-3x^2-x+4, x\geq 3. Assuming f(x) has an inverse g(x), find the value of g′(16)g'(16), given that f(4)=16.f(4)=16.
Practice: Tangent to the Inverse
Q:\textbf{Q:} Let f(x)=x3f(x)=x^3. Find the slope of the tangent line to f−1 at x=8=f(2)f^{-1}\text{ at }x=8=f(2).
Suppose f(x)=x3−3x2−x+4f(x)=x^3-3x^2-x+4 with x≥3x\geq 3. Assuming f(x)f(x) has an inverse g(x)g(x), find the value of g′(16)g'(16), given that f(4)=16f(4)=16 .
If f(x)=ex4f(x)=e^{x^4} find f−1(x)f^{-1}(x)and the derivative of f−1(e81)f^{-1}(e^{81}).
The function f(x)=x3+x+1f(x) = x^3 + x + 1 (where x>0x > 0) has an inverse, y=f−1(x)y=f^{-1}(x), on that interval. What is the slope of the curvey=f−1(x) y = f^{-1}(x) at the point (x,y)=(3,1)(x, y) = (3, 1)?
Derivatives: Inverse Functions
Suppose f(x)=x3−3x2−x+4f(x) = x^3-3x^2-x+4, x≥3x \geq 3. Assuming f(x)f(x) has an inverse g(x)g(x), find the value of g′(16)g'(16), given that f(4)=16f(4) = 16.
Derivative of inverse functions
Let f(x)=x3+4x−2. Find (f−1)′(3).f(x)=x^3+4x−2.\ Find\ (f^{−1})'(3).