# Product Rule

#### Jacob

##### New member
Good day, I stuck on a question and I need someone to check my working and guide me where I went wrong.

y = (2x + 7)^5 (10 - x)^7

u = (2x +7)^5 v = (10 - x)^7
u' = 5(2x +7)^4 v' = 7(10 - x)^6

Formula: Uv' + Vu'
(2x+7)^5 [7( 10 - x )^6] + (10 - x)^7 [5( 2x + 7)^4]
(2x + 7)^4 (10 - x)^6 [ Here is where I stuck ] my lecturer told me to look for the lowest power and assign it to the first equation which is 4 and highest is 6 I assume I did that right but IDK how to solve the inside the bracket. I'm not looking for an answer but also looking for an explanation of how to solve.

#### MarkFL

##### Super Moderator
Staff member
The first error I see is that you are not applying the chain rule when computing the derivatives.

#### topsquark

##### Full Member
After you get done with that, factor factor factor!

-Dan

#### Jacob

##### New member
The first error I see is that you are not applying the chain rule when computing the derivatives.
u' = 10(2x + 7)^5 and v' = -7(10 - x)^6, guess I get it right now

(2x + 7)^4 (10 - x)^6 [ ] but still how do I plug it correctly? I'm not very on this step

#### MarkFL

##### Super Moderator
Staff member
You want:

$$\displaystyle u'=10(2x+7)^4$$

I know that's just a typo because you've demonstrated already you understand the power rule. And so, given:

$$\displaystyle y=(2x + 7)^5(10 - x)^7$$

then:

$$\displaystyle y'=(2x + 7)^5\cdot7(10 - x)^6(-1)+5(2x + 7)^4(2)(10 - x)^7=-7(2x + 7)^5(10 - x)^6+10(2x + 7)^4(10 - x)^7$$

Now, you've correctly identified the greatest common factor $$(2x + 7)^4 (10 - x)^6$$, and so you will factor that out, and subtract the exponent on each factor of the GCF from the corresponding terms in what remains:

$$\displaystyle y=(2x + 7)^4 (10 - x)^6\left(10(2x + 7)^{4-4}(10 - x)^{7-6}-7(2x + 7)^{5-4}(10 - x)^{6-6}\right)$$

Can you proceed?

#### Jacob

##### New member
$$\displaystyle y=(2x + 7)^4 (10 - x)^6\left(10(2x + 7)^{4-4}(10 - x)^{7-6}-7(2x + 7)^{5-4}(10 - x)^{6-6}\right)$$

Can you proceed?
Yes. The answer will be (2x + 7)^4 (10-x)^6 [-24x + 51]. Thank you so much for the explanation.

#### Jomo

##### Elite Member
Looks good to me.