求导去积分符号，积分去求导符号

一、题目

$$f^{(n)}(0) = ?$$

二、解析

\textcolor{yellow}{ \begin{aligned} f(0) \\ \\ & = \int_{0}^{x} \mathrm{e}^{ -f(t) } \mathrm{~d} t \Bigg|_{x = 0} \\ \\ & = 0 \end{aligned} }

\begin{aligned} & f(\textcolor{springgreen}{x}) = \int_{0}^{\textcolor{springgreen}{x}} \mathrm{e}^{ -f(\textcolor{orangered}{t}) } \mathrm{ ~d } \textcolor{orangered}{t} \\ \\ \Rightarrow & \ \text{等号两边同时对} \ \textcolor{springgreen}{x} \ \text{求导} \\ \\ \Rightarrow & \ f^{ \prime }(\textcolor{springgreen}{x}) = \mathrm{e}^{ -f(\textcolor{springgreen}{x}) } \\ \\ \Rightarrow & \ \textcolor{pink}{e^{ f(x) }} \cdot f^{ \prime }(\textcolor{springgreen}{x}) = \textcolor{pink}{e^{ f(x) }} \cdot \mathrm{e}^{ -f(\textcolor{springgreen}{x}) } \\ \\ \Rightarrow & \ e^{ f(\textcolor{springgreen}{x}) } \cdot f ^ { \prime } (\textcolor{springgreen}{x} ) = 1 \\ \\ \Rightarrow & \ \text{等号两边同时对} \ \textcolor{springgreen}{x} \ \text{积分} \\ \\ \Rightarrow & \ \int e^{ f(\textcolor{springgreen}{x}) } \cdot f ^ { \prime } (\textcolor{springgreen}{x} ) \mathrm{~d} x = \int 1 \mathrm{~d} x \\ \\ \Rightarrow & \ \mathrm { e } ^ { f ( \textcolor{springgreen}{x} ) } = \textcolor{springgreen}{x} + C \\ \\ \Rightarrow & \ \textcolor{yellow}{ f(0) = 0 \rightarrow \mathrm{e}^{0} = C \rightarrow C = 1 } \\ \\ \Rightarrow & \ \mathrm { e } ^ { f ( x ) } = x + 1 \\ \\ \Rightarrow & \ \textcolor{springgreen}{ f ( x ) = \ln ( x + 1 ) } \end{aligned}

\textcolor{yellow}{ \begin{aligned} \ln (1+x) & = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{n+1} \\ \\ & = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} x^{n} \end{aligned} }

\begin{aligned} & \frac{f^{(n)} (x_{0})}{n!} \cdot (x – x_{0})^{n} \\ \\ \Rightarrow & \ x_{0} = 0 \\ \\ \Rightarrow & \ \frac{f^{(n)}(0)}{n!} \cdot x^{n} \end{aligned}

\begin{aligned} & \frac{f^{(n)}(0)}{n!} \cdot x^{n} = \frac{(-1)^{n-1}}{n} x^{n} \\ \\ \Rightarrow & \ \frac{f^{(n)}(0)}{n!} = \frac{(-1)^{n-1}}{n} \\ \\ \Rightarrow & \ f^{(n)}(0) = \frac{(-1)^{n-1}}{n} \cdot \textcolor{orangered}{n !} \\ \\ \Rightarrow & \ f^{(n)}(0) = (-1)^{n-1} \cdot \textcolor{orangered}{ \frac{n (n-1) !}{n} } \\ \\ \Rightarrow & \ \textcolor{springgreen}{\boldsymbol{ f^{(n)}(0) = (-1)^{n-1}(n-1)! }} \end{aligned}