Exercise 1, § 2

Functions of a Real Variable, III, 125

Let $\mathbf{f}$ be a vector function that is $n$ times differentiable on an interval $I \subset \mathbf{R}$ . Prove the formula

D^n \mathbf{f}(e^x) = \sum_{m=1}^n \frac{a_m}{m!}\ e^{mx}\ \mathbf{f}^{(m)}(e^x)

at every point

x

such that

e^x \in I

, where the coefficient

a_m

can be expressed as

a_m = \sum_{p=0}^m (-1)^p \binom{m}{p} (m-p)^n

(method of I, p. 41, exerc. 7, using the Taylor expansion for

e^x

Solution

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