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

Junior 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.