Question

If `f(x)= csc7 x` and `g(x) = e^(1 +3x )`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

`d/(dx)f(g(x))=-21e^((1+3x))csc(7e^(1+3x))cot(7e^(1+3x))`

Explanation:

As `f(x)=csc7x` and `g(x)=e^(1+3x)`

`f(g(x))=csc(7e^(1+3x))`

The chain in the derivative of `f(g(x))` is in fact longer.

We have `f(x)=csc(u(x))`, `u(g(x))=7g(x)`, `g(x)=e^(v(x))` and `v(x)=1+3x`.

Hence `d/(dx)f(g(x))=(df)/(du)xx(du)/(dg)xx(dg)/(dv)xx(dv)/(dx)`

= `-cscucotuxx7xxe^vxx3`

= `-21e^((1+3x))csc(7e^(1+3x))cot(7e^(1+3x))`