Question

Use the formula in Exercise 83.

84. If \(f\left( 4 \right) = 5\) and \(f'\left( 4 \right) = \frac{2}{3}\), find \({\left( {{f^{ - 1}}} \right)^\prime }\left( 5 \right)\).

Short answer

The value is \(\frac{3}{2}\).

Step-by-step solution

The derivative of the inverse function

If \(f\left( x \right)\) be a differentiable and invertible function, then the derivative of \({f^{ - 1}}\) can be obtained using the formula: \({\left( {{f^{ - 1}}} \right)^\prime }\left( x \right) = \frac{1}{{f'\left( {{f^{ - 1}}\left( x \right)} \right)}}\).

The derivative of given inverse

It is given that \(f\left( 4 \right) = 5\) that implies \({f^{ - 1}}\left( 5 \right) = 4\).

Substitute \(x = 5\) in the above formula:

\begin{aligned} & {{\left( {{f}^{-1}} \right)}^{\prime }}\left( 5 \right)=\frac{1}{{f}'\left( {{f}^{-1}}\left( 5 \right) \right)} \\ & {{\left( {{f}^{-1}} \right)}^{\prime }}\left( 5 \right)=\frac{1}{{f}'\left( 4 \right)} \\ & {{\left( {{f}^{-1}} \right)}^{\prime }}\left( 5 \right)=\frac{1}{{2}/{3}\;} \\ & {{\left( {{f}^{-1}} \right)}^{\prime }}\left( 5 \right)=\frac{3}{2} \\\end{aligned}

Hence, \({\left( {{f^{ - 1}}} \right)^\prime }\left( 5 \right) = \frac{3}{2}\).