Derivative Homework: y=x4(2x-5)6

[polak333]

Homework Statement

y =x⁴(2x-5)⁶

Homework Equations

Product Rule & Power of a Function Rule

The Attempt at a Solution

y = x⁴(2x-5)⁶
y' = 4(x)³(1)(2x-5)⁶ + x⁴(6)(2x-5)⁵(2)
y' = 4x³(2x-5)⁶ + 12x⁴(2x-5)⁵

The answer is:
20x³(2x-5)⁵(x-1)

No idea where they get the 20x³ or the (x-1). If I were to factor my answer, I still wouldn't get that, I think.

 

[Mark44]

Your answer is correct but unfactored. Pull the greatest common factor (GCF) out of the two separate terms, as shown below.
4x³(2x - 5)⁶ + 12x⁴(2x - 5)⁵
= 4x³(2x - 5)⁵(2x - 5 + 3x)
= 4x³(2x - 5)⁵(5x - 5)
= 5*4x³(2x - 5)⁵(x - 1)
= 20x³(2x - 5)⁵(x - 1)

 

[polak333]

Thanks!

Can you help me out with one more?

y = (1-x²)³ (6+2x)^-3
y' = 3 (1-x²)² (-2x)(6+2x)^-3 + (1-x²)³(-3)(6+2x)^-4(2)
y' = -6x (1-x²)²(6+2x)^-3 - 6(1-x²)³(6+2x)^-4

Not exactly sure what to do with this.

I could possibly:
y' = -6 (1-x²)(6+2x)^-3[x-(1-x²)(6+2x)^-1]

or should I put the (6+2x) on the bottom:
y' = -6x(1-x²)² - 6(1-x²)³
...---------- .. ----------
...(6+2x)³ ... (6+2x)⁴


The answer is:

-6(1-x²)²(x²+6x+1)
-------------------
...(6+2x)⁴

 

[Mark44]

The common factor is 6(1 - x²)²(6 + 2x)^-4. Pull that out and then combine what's left.

 

[polak333]

y = (1-x²)³ (6+2x)^-3
y' = 3 (1-x²)² (-2x)(6+2x)^-3 + (1-x2)3(-3)(6+2x)^-4(2)
y' = -6x (1-x²)²(6+2x)^-3 - 6(1-x2)3(6+2x)^-4

That's kind of the problem. I'm not sure how to take out < -6(1 - x²)²(6 + 2x)^-4 >.

In the 3rd line, there is a (6+2x)^-3, how do you take out (6+2x)^-4? Does the power become a positive, and therefore: (6+2x)¹ which is just (6+2x)?

 

[polak333]

Nevermind. I got it.

Thanks for the help!