Product rule. what did I do wrong?

[tony873004]

It's been too long since I've had an algebra class, so I start getting into trouble as these calculus questions rely more and more on algebra.

$$\begin{array}{l} y = (4x - 5)^4 \,(3x + 1)^5 \\ {\rm{Find the first derivative}}{\rm{. Simplify if possible (i}}{\rm{.e}}{\rm{. factor)}}{\rm{. Use the product rule}}{\rm{.}} \\ \left( {(4x - 5)^4 } \right)^\prime \left( {(3x + 1)^5 } \right) + \left( {(4x - 5)^4 } \right)\left( {(3x + 1)^5 } \right)^\prime \\ \\ \left( {4(4x - 5)^3 (4x - 5)'} \right)\left( {(3x + 1)^5 } \right) + \left( {(4x - 5)^4 } \right)\left( {5(3x + 1)^4 (3x + 1)'} \right) \\ \\ 4(4x - 5)^3 (4)(3x + 1)^5 + (4x - 5)^4 \,\,5(3x + 1)^4 (3) \\ \\ 16(4x - 5)^3 (3x + 1)^5 + 15(4x - 5)^4 (3x + 1)^4 \\ \end{array}$$
I only got 1 point out of 2 on this question. But when I graph it to check the answer, my formula seems to give me the correct slope at all points on the original function. Was the teacher expecting me to simplify this answer?

 

[dav2008]

Yes you have the correct answer. I guess the teacher wanted you to expand, which is silly because it's an 8th degree polynomial with coefficients in the millions.

Maybe he wanted you to factor it even more.

Edit: Well after reading the jibberish you wrote in tex the problem said to factor the final answer.

 

[tony873004]

Perhaps the teacher wanted this:
$$\left( {4x - 5} \right)^3 \left( {3x + 1} \right)^4 \left( {16(3x + 1) + 15(4x - 5)} \right)$$
Should I try to factor out the 15 and the 16, which would eliminate one of them, and leave me with a fraction? That would look ugly. Do you suppose the teacher would have wanted this?

 

[tony873004]

Edit: Well after reading the jibberish you wrote in tex the problem said to factor the final answer.

Sorry about that. Tex ignored my spaces. From now on, I'll put my text outside the TEX.

 

[dav2008]

Well if you just expand that last term you end up with $$\left( {4x - 5} \right)^3 \left( {3x + 1} \right)^4 \left( {108x-59} \right)$$

Chances are that's what he was looking for. When in doubt just ask him what he was looking for.

 

[tony873004]

Thanks dav2008. His office hours are impossible for me because of work. And there's always a line of students asking q's after class.

Stay tuned for more questions. I'm going over my last 2 tests and trying to figure out all the problems I missed. I might see these on the final!