Question

设f(x)可导,计算函数=f(ex+x)的导数dy/dx

Answer 1

y=f(e^x + x)

记：u(x)=e^x + x

y=f(u)

由复合函数求导规则：

dy/dx = df/du du/dx

dy/dx = f '(u) (e^x + 1)

举例：设：f(u)=sin² u u=e^x + x 即：y=sin²(e^x + x)

f '(u)=df/du=2sinucosu=sin(2u)

du/dx=e^x + 1

dy/dx=sin(2u) (e^x + 1)

=sin[2(e^x+x)] (e^x +1)