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 $$g^\prime (2) =\dfrac{1}{f^\prime (1) }=\dfrac{1}{6}$$