Exercise 4, § 3

Functions of a Real Variable, I, 40

Let $\mathbf{f}$ be a vector function which is $n$ times differentiable on an interval $I \subset \mathbf{R}$ . Show that for $1/x \in I$ one has identically

\frac{1}{x^{n+1}} \mathbf{f}^{(n)} \left( \frac{1}{x} \right) = (-1)^n D^n \left( x^{n-1} \mathbf{f} \left( \frac{1}{x} \right) \right)

(argue inductively on $n$ ).

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