Question

What is the derivative of `y=2(x^(3)-1)(3x^(2)+1)^4`?

Answer 1

`y^' = 6x(3x^2 + 1) * (11x^3 +x - 8)`

Explanation:

You can differentiate this function by using the product rule and the chain rule.

If you write `y` as

`y = f(x) * g(x)`

then you can say that the derivative of `y` is

`d/dx(y) = 2 * [[d/dx(f(x))] * g(x) + f(x) * d/dx(g(x))]`

In your case, `f(x) = (x^3-1)` and `g(x) = (3x^2+1)^4`.

`y^' = 2 [d/dx(x^3-1) * (3x^2 + 1)^4 + (x^3-1) * d/dx(3x^2+1)^4]`

To determine `d/dx(g(x))`, use the chain rule for

• `g(u) = u^4`, with `u = 3x^2 + 1`

`d/dx(g(u)) = d/(du)(u^4) * d/dx(u)`

`d/dx(g(u)) = 4u^3 * d/dx(3x^2 + 1)`

`d/dx(3x^2+1) = 4(3x^2+1)^3 * 6x`

Your target derivative will thus be

`y^' = 2 * [3x^2 * (3x^2+1)^4 + (x^3-1) * 24x * (3x^2+1)^3]`

`y^' = 2 * 3x * (3x^2 + 1) * [x * (3x^2 + 1) + (x^3-1) * 8]`

`y^' = 6x(3x^2 + 1) * (3x^3 + x + 8x^3 - 8)`

`y^' = color(green)(6x(3x^2 + 1) * (11x^3 +x - 8))`