Math help please?

Question

Use the product rule to find the derivative.

f(x) = (x^2 - 2x + 2)(2x^3 - x^2 + 5)


if you could explain this problem step by step it would help thank you.

Answer 1

f'(x) = (x^2 - 2x + 2)'(2x^3 - x^2 + 5) + (x^2 - 2x + 2)(2x^3 - x^2 + 5)' = (2x-2)(2x^3 - x^2 + 5) + (x^2 - 2x + 2)(6x^2 - 2x)

Answer 2

Hi,
Well, the product rule says the derivative of a product is this:
(first term)(derivative of second term) + (second term)(derivative of first term)
So, let's see how that works using (x^2 - 2x + 2) as the first term and (2x^3 - x^2 + 5) as the second term.
To simplify it a little, maybe, let's take the derivatives of the first and second terms:
d(x^2 - 2x + 2)/dx =2x -2

d(2x^3 - x^2 + 5)/dx = 6x² -2x
That's just using x^n = nx^(n-1).
Now, let's put it all together:
(first term)(derivative of second term) + (second term)(derivative of first term)
(x^2 - 2x + 2)(6x²-2x) +(2x^3 - x^2 + 5)(2x-2)
Now, hold on. This is going to get tedious.
6x^4 - 12x^3 +12x^2 + (-2x^3+4x^2-4x) + 4x^4-2x^3+10x +(-4x^3+2x^2-10)
(6x^4 -14x^3 +16x^2 -4x) +(4x^4 -6x^3+2x^2 +10x -10)
10x^4 -20x^3 +18x^2 +6x - 10

Frankly, I've never really given my students this type of busy work. I believe there are more important things to learn. But that's it anyway.

Hope this helps.
FE

Answer 3

There are two products here, so we take the derivative of the first product and multiply it by the second, and then add to that the first product multiplied by the derivative of the second product.

Derivative of the first is 2x-2. Multiply that with the second and you get (2x-2)(2x^3-x^2+5).

Derivative of the second is 6x^2-2x. Multiply that with the first and you get (x^2-2x+2)(6x^2-2x).

Add them together to get

(2x-2)(2x^3-x^2+5)+(x^2-2x+2)(6x^2-2x)