Exercise 2, § 2

Let $f$ be a real function that is $n$ times differentiable at a point $x$, and $g$ a vector function $n$ times differentiable at the point $f(x)$. If one puts $D^n(g(f(x))) = \sum_{k=1}^n g^{(k)}(f(x)) u_k(x)$, then $u_k$ depends on the function $f$ alone; deduce from this that $u_k(x)$ is the coefficient of $t^k$ in the expansion of $e^{-tf(x)} D^n(e^{tf(x)})$ (in terms of $t$).