Question

If `f(x)= cos5 x` and `g(x) = e^(3+4x )`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

Leibniz's notation can come in handy.

Explanation:

`f(x)=cos(5x)`

Let `g(x)=u`. Then the derivative:

`(f(g(x)))'=(f(u))'=(df(u))/dx=(df(u))/(dx)(du)/(du)=(df(u))/(du)(du)/(dx)=`

`=(dcos(5u))/(du)*(d(e^(3+4x)))/(dx)=`

`=-sin(5u)*(d(5u))/(du)*e^(3+4x)(d(3+4x))/(dx)=`

`=-sin(5u)*5*e^(3+4x)*4=`

`=-20sin(5u)*e^(3+4x)`