what is the necessary and sufficient condition $f^{g(x)}(x)$ to be differentiable at $x$

Question

Let $f$ and $g$ are real functions

1. can you give me a counter example to the statement

"if $f$ and $g$ are differentiable functions then $f^{g(x)}(x)$ is differentiable"

2. what is the necessary and sufficient condition $f^{g(x)}(x)$ to be differentiable at $x$

thanks.

Answer 1

$f^{g(x)} (x)$ is not well defined if $f$ takes negative values. So assume that $f$ is non-negative. Then $f^{g(x)} (x)$ at a point $x_0$ iff $f(x_0)>0$ . It is differentiable at all points iff $f(x)>0$ for all $x$. To prove these statements simply write $f^{g(x)} (x)$ as $e^{g(x) \ln f(x)}$.