Question

How do you differentiate `f(x)= (4x^5+5)^(1/2)` using the chain rule?

Answer 1

`{d[f(x)]}/{dx} = {10x^4}/{sqrt{4x^5 + 5}}`

Explanation:

`f(x) = (4x^5 + 5)^{1/2}`

`{d[f(x)]}/{dx} = d/dx[(4x^5 + 5)^{1/2}]`

let `u = 4x^5 + 5`

`{d[f(x)]}/{dx} = d/{du}[u^{1/2}]d/dx[4x^5 + 5]`

`{d[f(x)]}/{dx} = 1/2u^{-1/2} times 20x^4`

Substitute `u` back in

`{d[f(x)]}/{dx} = 1/2( 4x^5 + 5)^{-1/2} times 20x^4`

Simplify

`{d[f(x)]}/{dx} = {10x^4}/{sqrt{4x^5 + 5}}`

Answer 2

`f'(x)=(10x^4)/(4x^5+5)^(1/2)`

Explanation:

`"given "f(x)=g(h(x))" then"`

`f'(x)=g'(h(x))xxh'(x)larrcolor(blue)"chain rule"`

`rArrf'(x)=1/2(4x^5+5)^(-1/2)xxd/dx(4x^5+5)`

`"color(white)(rArrf'(x))=1/cancel(2)(4x^5+5)^(-1/2)xxcancel(20)^(10)x^4`

`color(white)(rArrf'(x))=(10x^4)/(4x^5+5)^(1/2)`