Product Rule

Jacob

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Jan 27, 2019
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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

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The first error I see is that you are not applying the chain rule when computing the derivatives. :)
 

topsquark

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Aug 27, 2012
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After you get done with that, factor factor factor!

-Dan
 

Jacob

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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

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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

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Jan 27, 2019
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\(\displaystyle y=(2x + 7)^4 (10 - x)^6\left(10(2x + 7)^{4-4}(10 - x)^{7-6}-7(2x + 7)^{5-4}(10 - x)^{6-6}\right)\)

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

Jomo

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Looks good to me.
 
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