Question

Determine the value of\({\left( {{f^{ - 1}}} \right)^\prime }(5)\).

Short answer

The value of \({\left( {{f^{ - 1}}} \right)^\prime }(5)\) is \(\frac{3}{2}\).

Step-by-step solution

Given function

The given function is, \(f(4) = 5\).

Concept of inverse function

If\(f\)is a one-to-one differentiable function with inverse function\({f^{ - 1}}\)and\({f^\prime }\left( {{f^{ - 1}}(a)} \right) \ne 0\), then the inverse function is differentiable at\(a\)and\({\left( {{f^{ - 1}}} \right)^\prime }(a) = \frac{1}{{{f^\prime }\left( {{f^{ - 1}}(a)} \right)}}\).

Use the theorem to find the value of \({\left( {{f^{ - 1}}} \right)^\prime }(5)\)

From the given information,\(f(4) = 5\).

Thus,\({f^{ - 1}}(5) = 4\).

With the help of the theorem If\(f\)is a one-to-one differentiable function with inverse function\({f^{ - 1}}\)and\({f^\prime }\left( {{f^{ - 1}}(a)} \right) \ne 0\), then the inverse function is differentiable at\(a\)and\({\left( {{f^{ - 1}}} \right)^\prime }(a) = \frac{1}{{{f^\prime }\left( {{f^{ - 1}}(a)} \right)}}\)obtain the value of\({\left( {{f^{ - 1}}} \right)^\prime }(2)\)as follows:

\(\begin{array}{c}{\left( {{f^{ - 1}}} \right)^\prime }(5) = \frac{1}{{{f^\prime }\left( {{f^{ - 1}}(5)} \right)}}\\ = \frac{1}{{{f^\prime }(4)}}\\ = \frac{1}{{\left( {\frac{2}{3}} \right)}}\\ = \frac{3}{2}\end{array}\)

Thus, the value of \({\left( {{f^{ - 1}}} \right)^\prime }(5)\) is \(\frac{3}{2}\).