Understanding the Chain Rule: A Derivative Problem Solution

[Phyzwizz]

I did a derivative problem, but my book says that my answer is wrong.

f(x)=x²(x-2)⁴

I didn't see much use in the chain rule so I used the product rule.

x²(4(x-2)³) + (x-2)⁴(2x)
=4x²(x-2)³ + 2x(x-2)⁴

The book says that instead of this, the answer is ...
x²(4(x-2)³(1)) + (x-2)⁴(2x) = 2x(x-2)³(3x-2)

So I guess where I'm really confused is in the manner with which the book simplified.

 

[HallsofIvy]

I did a derivative problem, but my book says that my answer is wrong.

f(x)=x²(x-2)⁴

I didn't see much use in the chain rule so I used the product rule.

x²(4(x-2)³) + (x-2)⁴(2x)
=4x²(x-2)³ + 2x(x-2)⁴

Well, you did use the chain rule when you said that the derivative of (x- 2)⁴ is 4(x- 2)³ (because the derivative of x- 2 is 1).

If you factor out 2x(x- 2)³ to get f'= 2x(x- 2)³(2x+ (x- 2))= 2x(x- 2)(3x- 2), exactly what your book says. Your answer and your book's answer are the same.

The book says that instead of this, the answer is ...
x²(4(x-2)³(1)) + (x-2)⁴(2x) = 2x(x-2)³(3x-2)

So I guess where I'm really confused is in the manner with which the book simplified.

 

[Phyzwizz]

Awesome thanks, I get it now. And I also now understand how ridiculous my book is in its mindless factoring.