问题

设 $f,g$ 均为 $3$ 阶可导函数, 求复合函数 $f(g)$ 的各阶导数.

$$\begin{array}{rl}(f(g){)}^{\prime }& =f^{\prime }(g)g^{\prime },\\ [f(g){]}^{″}& =[f^{\prime }(g)g^{\prime }{]}^{\prime }=[f^{\prime }(g){]}^{\prime }{g}^{\prime }+f^{\prime }(g)g^{″}=f^{″}(g)g^{\prime }{g}^{\prime }+f^{\prime }(g)g^{″},\end{array}$$
从而
$$\begin{array}{rl}[f(g){]}^{(3)}& =[f^{″}(g)g^{\prime }{g}^{\prime }+f^{\prime }(g)g^{″}{]}^{\prime }\\ & =f^{‴}(g)(g^{\prime }{)}^3+f^{″}(g)2g^{\prime }{g}^{″}+f^{″}(g)g^{\prime }{g}^{″}+f^{\prime }(g)g^{‴}\\ & =f^{(3)}(g)(g^{\prime }{)}^3+3f^{″}(g)g^{\prime }{g}^{″}+f^{\prime }(g)g^{(3)}.\end{array}$$

经过计算

$$[f(g){]}^{(4)}=f^{(4)}(g)(g^{\prime }{)}^4+6f^{(3)}(g^{\prime }{)}^2{g}^{″}+3f^{″}(g)(g^{″}{)}^2+4f^{″}(g)g^{\prime }{g}^{(3)}+f^{\prime }(g)g^{(4)},$$

$$\begin{array}{rl}[f(g){]}^{(5)}& =f^{(5)}(g)(g^{\prime }{)}^5+10f^{(4)}(g)(g^{\prime }{)}^3{g}^{″}+10f^{(3)}(g)(g^{\prime }{)}^2{g}^{(3)}+15f^{(3)}(g)g^{\prime }(g^{″}{)}^2\\ & +10f^{″}(g)g^{″}{g}^{(3)}+5f^{″}(g)g^{\prime }{g}^{(4)}+f^{\prime }(g)g^{(5)}.\end{array}$$