How do you differentiate y =2x^3(x^3 - 3)^4 y=2x3(x33)4 using the chain rule?

1 Answer
Jul 26, 2016

dy/dx = 6x^2(x^3 - 3)^4 +24x^5(x^3 - 3)^3dydx=6x2(x33)4+24x5(x33)3

Explanation:

This problem actually requires the application of both the product rule and the chain rule.

The product rule states that
d/dx f(x)g(x) = f'(x)g(x) + f(x)g'(x)

And the chain rule states that
d/dx f(g(x)) = f'(g(x))g'(x)

We'll start by applying the product rule:
dy/dx = 6x^2(x^3 - 3)^4 +2x^3(d/dx (x^3 - 3)^4)

Now we use the chain rule to differentiate the parenthetical term:
= 6x^2(x^3 - 3)^4 +2x^3(4(x^3 - 3)^3 * 3x^2)

We can simplify a bit to make things cleaner and combine terms:
= 6x^2(x^3 - 3)^4 +24x^5(x^3 - 3)^3