Question

Find F'(a) and G'(a) ?

Let `F(x) = f(x^4) and G(x) = (f(x))^4` ?
You also know that `a^3=3, f(a) = 3, f'(a) = 9, f'(a^4) = 11`

Answer 1

`F'(a) = 88`
`G'(a) = 1215`

Explanation:

We have:

`F(x) = f(x^4)`
`G(x)=f(x)^4`

Along with:

`a^3=3, f(a)=3, f'(a)=9, f'(a^4)=11`

We can differentiate these function using the chain rule:

`dy/dx=dy/(du)*(du)/dx`

which I will show explicitly using a substitution.

Let `u=x^4 => (du)/dx = 4x^3` so that `F(x) = f(u)`

Then we have;

`F'(x) = (df)/dx`
`" " = (df)/(du) * (du)/dx " "` (chain rule)
`" " = f'(u) * 4x^3`
`" " = 4x^3 \ f'(x^4)`

Similarly we can do the same for `G(x)` and we get:

`G'(x) = 4 \ f(x)^3 \ f'(x)`

So when `x=a` we get the following:

`F'(a) = 4a^3 \ f'(a^4)`
`" " = 4 * 4 * 11`
`" " = 88`

And,

`G'(a) = 4 \ f(a)^3 \ f'(a)`
`" " = 5 * 3^3 * 9`
`" " = 1215`