Question

If `f(x)= cos 4 x` and `g(x) = 2 x`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

`-8sin(8x)`

Explanation:

The chain rule is stated as:

`color(blue)((f(g(x)))'=f'(g(x))*g'(x))`

Let's find the derivative of `f(x)` and `g(x)`

`f(x)=cos(4x)`
`f(x)=cos(u(x))`

We have to apply chain rule on `f(x)`
Knowing that `(cos(u(x))'=u'(x)*(cos'(u(x))`

Let `u(x)=4x`
`u'(x)=4`

`f'(x)=u'(x)*cos'(u(x))`

`color(blue)(f'(x)=4*(-sin(4x))`

`g(x)=2x`
`color(blue)(g'(x)=2)`

Substituting the values on the property above:

`color(blue)((f(g(x)))'=f'(g(x))*g'(x))`

`(f(g(x)))'=4(-sin(4*(g(x)))*2`
`(f(g(x)))'=4(-sin(4*2x))*2`

`(f(g(x)))'=-8sin(8x)`