Derivative – Inverse Function

The derivative of an inverse function allows us to find the slope of an inverse function without explicitly solving for the full inverse equation. The key formula is \(\left(f^{-1}\right)'(a)=\frac{1}{f'(f^{-1}(a))}\), which uses the derivative of the original function evaluated at the input that gives output \(a\). These problems focus on finding \(f^{-1}(a)\), evaluating \(f'(x)\), and then applying the inverse function derivative formula.

 

Derivative of the Inverse Function
\(\displaystyle \left(f^{-1}\right)'(a)=\frac{1}{f’ \left( f^{-1}(a) \right)}\)

 

Practice Problems

\(\textbf{1)}\) Determine \(\left(f^{-1}\right)'(8)\) for \(f(x) = x^3\).

 

\(\textbf{2)}\) Determine \(\left(f^{-1}\right)'(6)\) for \(f(x) = 15-x^2, x\ge 0\).

 

\(\textbf{3)}\) Determine \(\left(f^{-1}\right)'(4)\) for \(f(x) = \sqrt{x}\).

 

\(\textbf{4)}\) Determine \(\left(f^{-1}\right)'(9)\) for \(f(x) = x^2, x\ge 0\).

 

\(\textbf{5)}\) Determine \(\left(f^{-1}\right)'(1)\) for \(f(x) = x^2-4x+4, x\ge 2\).

 

\(\textbf{6)}\) Determine \(\left(f^{-1}\right)'(27)\) for \(f(x)=x^3\).

 

\(\textbf{7)}\) Determine \(\left(f^{-1}\right)'(16)\) for \(f(x)=x^2,\;x\ge0\).

 

\(\textbf{8)}\) Determine \(\left(f^{-1}\right)'(25)\) for \(f(x)=\sqrt{x}\).

 

\(\textbf{9)}\) Determine \(\left(f^{-1}\right)'(1)\) for \(f(x)=e^x\).

 

\(\textbf{10)}\) Determine \(\left(f^{-1}\right)'(e^2)\) for \(f(x)=e^x\).

 

\(\textbf{11)}\) Determine \(\left(f^{-1}\right)'(3)\) for \(f(x)=\ln{x}\).

 

\(\textbf{12)}\) Determine \(\left(f^{-1}\right)'(0)\) for \(f(x)=x^3+x\).

 

\(\textbf{13)}\) Determine \(\left(f^{-1}\right)'(5)\) for \(f(x)=x+4\).

 

\(\textbf{14)}\) Determine \(\left(f^{-1}\right)'(9)\) for \(f(x)=\sqrt{x+7}\).

 

\(\textbf{15)}\) Determine \(\left(f^{-1}\right)'(2)\) for \(f(x)=x^3+1\).

 

\(\textbf{16)}\) Find the derivative of the inverse function of \(f(x)=x^5\) at \(a=32\).

 

\(\textbf{17)}\) Find the derivative of the inverse function of \(f(x)=x^3-8\) at \(a=0\).

 

\(\textbf{18)}\) Find the derivative of the inverse function of \(f(x)=2x+1\) at \(a=9\).

 

\(\textbf{19)}\) Find the derivative of the inverse function of \(f(x)=x^2+1,\;x\ge0\) at \(a=10\).

 

\(\textbf{20)}\) Find the derivative of the inverse function of \(f(x)=\sqrt{x-5}\) at \(a=4\).

 

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of Graph of First and Second Derivatives
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

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