Question:
If \(f\) is a differentiable function such that
\[
f(1)=8
\]
and
\[
f'(1)=\frac18.
\]
If \(f\) is invertible and \(g=f^{-1}\), then
If \(f\) is a differentiable function such that
\[
f(1)=8
\]
and
\[
f'(1)=\frac18.
\]
If \(f\) is invertible and \(g=f^{-1}\), then
Show Hint
Remember the inverse function derivative formula:
\[
(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}.
\]
First find \(f^{-1}(a)\), then substitute into the derivative of the original function.
Updated On: Jul 29, 2026
- \(g'(1)=8\)
- \(g'(1)=\dfrac18\)
- \(g'(8)=8\)
- \(g'(8)=\dfrac18\)
Show Solution
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The Correct Option is C
Solution and Explanation
Concept:
If \(g=f^{-1}\), then the derivative of the inverse function is
\[
g'(y)=\frac{1}{f'(x)},
\]
where
\[
y=f(x).
\]
Equivalently,
\[
(f^{-1})'(a)
=
\frac{1}{f'(f^{-1}(a))}.
\]
Step 1: Find \(g(8)\). Given, \[ f(1)=8. \] Since \(g=f^{-1}\), \[ g(8)=1. \]
Step 2: Apply the inverse function derivative formula. \[ g'(8) = \frac{1}{f'(g(8))}. \] Substituting \(g(8)=1\), \[ g'(8) = \frac{1}{f'(1)}. \] Given \[ f'(1)=\frac18, \] therefore \[ g'(8) = \frac{1}{\frac18} = 8. \] Hence, \[ \boxed{g'(8)=8} \] \[ \boxed{\text{Answer = (C)}} \]
Step 1: Find \(g(8)\). Given, \[ f(1)=8. \] Since \(g=f^{-1}\), \[ g(8)=1. \]
Step 2: Apply the inverse function derivative formula. \[ g'(8) = \frac{1}{f'(g(8))}. \] Substituting \(g(8)=1\), \[ g'(8) = \frac{1}{f'(1)}. \] Given \[ f'(1)=\frac18, \] therefore \[ g'(8) = \frac{1}{\frac18} = 8. \] Hence, \[ \boxed{g'(8)=8} \] \[ \boxed{\text{Answer = (C)}} \]
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