i need to find the derivative using chain rule:

(x^2 + 2x - 6)^2 (1-x^3)^2

i got the answer from a site but the problem is i cannot get my work to match up with the answer, i don't know what im doing wrong.

answer: 10x^9 + 36x^8 − 64x^7 − 182x^6 + 168x^5 + 80x^4 + 196x^3 − 204x^2 − 16x−24

my work:
f'(x) = [2(x^2 + 2x - 6)*(2x + 2)]*(1-x^3)^2 + [2(1-x^3)*(3x^2)]*(x^2 +2x-6)^2

f'(x)=2(2x^3 +6x^2 -8x-12)*(1-x^3)^2 +[2(3x^2 -3x^5)]*(x^2 +2x-6)(x^2 +2x-6)

f'(x)= (4x^3 +12x^2 -16x-24)*(1-2x^3 +x^6) +(6x^2 -6x^5)*(x^4 +4x^3 -8x^2 -24x+36)
f'(x)=4x^9 +12x^8 -16x^7 -32x^6 -24x^5 +32x^4 +52x^3 +12x^2 -16x-24)+(-6x^9 -24x^8 +48x^7 -138x^6 -198x^5 -48x^4 -144x^3 +216x^2)

f'(x)=-2x^9 -12x^8 +32x^7 -170x^6 +222x^5 -16x^4 -92x^3 +228x^2 -16x -24

its 1am and i could have made some arithmetic errors but im fairly certain there accurate for the most part

well, maybe they're as accurate as your spelling...

Your very first line is in error: should be -3x^2 toward the end.

Fix that, and everything's ok.

If you visit wolframalpha.com and enter

derivative (x^2 + 2x - 6)^2 (1-x^3)^2

you will get several different ways of writing it.

If you enter

your corrected first step:

[2(x^2 + 2x - 6)*(2x + 2)]*(1-x^3)^2 + [2(1-x^3)*(-3x^2)]*(x^2 +2x-6)^2

you will get the same answer.

Devil's in the details, as usual. Unless otherwise instructed, I'd probably not take it all the way to a single expanded expression.

Y=2x^3-16x^2+64x+1

To find the derivative using the chain rule, you need to follow these steps:

Step 1: Identify the composition of functions within the expression. In this case, we have two compositions: (x^2 + 2x - 6) and (1 - x^3).

Step 2: Differentiate each composition separately. Let's denote the first composition as u and the second composition as v.

- For u = (x^2 + 2x - 6), the derivative du/dx can be found by applying the power rule and the sum rule, giving us du/dx = 2x + 2.

- For v = (1 - x^3), the derivative dv/dx can be found by applying the power rule and the negative rule, giving us dv/dx = -3x^2.

Step 3: Calculate the overall derivative using the chain rule formula: (f(g(x)))' = f'(g(x)) * g'(x).

- The overall derivative of the expression would be: f'(x) = u^2 * du/dx * v^2 * dv/dx.

- Substituting the values we found earlier, we have f'(x) = (x^2 + 2x - 6)^2 * (2x + 2) * (1 - x^3)^2 * (-3x^2).

Step 4: Simplify the expression using algebraic manipulations and distribute the terms.

When simplifying your work, it looks like you made some arithmetic errors. Here is the corrected version:

f'(x) = (x^2 + 2x - 6)^2 * (2x + 2) * (1 - x^3)^2 * (-3x^2)
= (4x^2 + 4x - 12) * (1 - x^3)^2 * (-3x^2)
= -12x^2(x^2 + x - 3) * (1 - x^3)^2

You can go ahead and expand this further to get the final answer.