设 $f,g$ 均为 $3$ 阶可导函数, 求复合函数 $f(g)$ 的各阶导数.
设 $f,g$ 均为 $3$ 阶可导函数, 求复合函数 $f(g)$ 的各阶导数.
\[
\begin{aligned}
(f(g))'&=f'(g)g',\\
[f(g)]''&=[f'(g)g']'=[f'(g)]'g'+f'(g)g''=f''(g)g'g'+f'(g)g'',
\end{aligned}
\]
从而
\[
\begin{split}
[f(g)]^{(3)}&=[f''(g)g'g'+f'(g)g'']'\\
&=f'''(g)(g')^3+f''(g)2g'g''+f''(g)g'g''+f'(g)g'''\\
&=f^{(3)}(g)(g')^3+3f''(g)g'g''+f'(g)g^{(3)}.
\end{split}
\]
经过计算
\[
[f(g)]^{(4)}=f^{(4)}(g)(g')^4+6f^{(3)}(g')^2g''+3f''(g)(g'')^2+4f''(g)g'g^{(3)}+f'(g)g^{(4)},
\]
\[
\begin{split}
[f(g)]^{(5)}&=f^{(5)}(g)(g')^5+10f^{(4)}(g)(g')^3 g''+10f^{(3)}(g)(g')^2 g^{(3)}+15f^{(3)}(g) g'(g'')^2\\
&\quad+10f''(g) g'' g^{(3)}+5f''(g) g' g^{(4)}+f'(g)g^{(5)}.
\end{split}
\]