Question

What is the derivative of `y=(x^2lnx)^4`?

Answer 1

`y^' = 4x^7 * ln^3(x) * (2ln(x) + 1)`

Explanation:

Notice that you can simplify your function to get

`y = (x^2)^4 * (lnx)^4`

`y = x^8 * ln^4(x)`

To differentiate this function you can use the product rule and the chain rule for `u^4`, with `u = ln(x)`.

`d/dx(y) = [d/dx(x^8)] * ln^4(x) + x^8 * d/dx(ln^4(x))`

`y^' = 8x^7 * ln^4(x) + x^8 * ([d/(du)(u^4)] * d/dx(u))`

`y^' = 8x^7 * ln^4(x) + x^color(red)(cancel(color(black)(8))) * 4 ln^3(x) * 1/color(red)(cancel(color(black)(x)))`

`y^' = 8x^7 * ln^4(x) + 4x^7 * ln^3(x)`

This can be simplified to give

`y^' = color(green)(4x^7 * ln^3(x) * (2ln(x) + 1))`