Finding the Derivative of an Inverse Function

The function \(f( x) =x^{5}+x\) has an inverse function \(g\). Find \(g^\prime (2) \).

Solution Using (7) with \(y_{0}=2\), we get \[ g^\prime (2) =\dfrac{1}{f^\prime (x_{0})}\qquad \hbox{where}\ 2=f( x_{0}) \]

A solution of the equation \[ f( x_{0}) =x_{0}^{5}+x_{0}=2 \] is \(x_{0}=1\). Since \(f^\prime ( x) =5x^{4}+1\), then \( f^\prime (x_{0})= f^\prime (1) =6\) and \[ g^\prime (2) =\dfrac{1}{f^\prime (1) }=\dfrac{1}{6} \]