Question

What is the derivative of `f(x)= x^(4/5) (x-5)^2`?

Answer 1

`(4(x-5)^2)/(5x^(1/5))+x^(4/5)(2x-10)`

Explanation:

Product rule states that:

`d/dxcolor(red)(f(x))color(blue)(g(x))=color(darkred)(f'(x))color(blue)(g(x))+color(red)(f(x))color(darkblue)(g'(x))`

Therefore,

`d/dxcolor(white).color(red)(x^(4/5))color(blue)((x-5)^2) = color(darkred)(4/5x^(-1/5))color(blue)((x-5)^2)+color(red)(x^(4/5))color(darkblue)(d/dx(x-5)^2)`

Everything is good up until we have to differentiate `(x-5)^2`. To do this, we need chain rule, which states that:

`d/dxcolor(orange)f(color(lightgreen)g(x)) = color(brown)(f'(color(lightgreen)g(x))*color(green)(g'(x))`

Therefore,

`d/dxcolor(orange)((color(lightgreen)(x-5))^2) = color(brown)(2(color(lightgreen)(x-5))*color(green)1) = 2x-10`

So our final derivative is:

`d/dxcolor(white).color(red)(x^(4/5))color(blue)((x-5)^2) = color(darkred)(4/5x^(-1/5))color(blue)((x-5)^2)+color(red)(x^(4/5))color(darkblue)((2x-10))`

And if we want to simplify it further, we get:

`d/dx color(white). x^(4/5)(x-5)^2=(4(x-5)^2)/(5x^(1/5))+x^(4/5)(2x-10)`

Final Answer