Question

Evaluate dy/dx at x=2.
y=(x+1)(x^2+2)(x^3+2)

Answer 1

There is my work. I got 356, but apparently the answer is 396. Where did I go wrong?

Answer 2

It's hard to tell where you were differentiating; it looks like you were using product rule.
What was the final expression for the derivative before you plugged in values?

Answer 3

That would be 15x^4+8x^3+6x^2+12x+4

Answer 4

Ok, it looks like you mixed in some chain rule with your product rule; maybe try just simplifying the whole thing by multiplying it all together, so you don't have to use product rule at all.
The derivative I got was
\[6x^5+5x^4+8x^3+12x^2+4x+4\]

Answer 5

Ah. At what point in the problem did you multiply everything together?

Answer 6

Very first step.

Answer 7

Oh. So derivative is not necessary at all? Doesn't evaluate at dy/dx mean to take the derivative?

Answer 8

Yes, you take the derivative, but if you simplify the expression first then you don't need to use product rule to differentiate.

Answer 9

Ah, I see how you did it. So you just multiplied it all out and then applied the power rule. In instances where you are asked to take the derivative of three functions like this, is that usually the best way to solve it?

Answer 10

It depends on my mood. I usually only use the product rule if I have to. If it's easy enough to simplify so that I don't have to use the product rule, then I will.
Sometimes the product rule is a lot quicker - seemed like a chore in this case, though.

Answer 11

And using the product rule also seemed wrong in this case. I didn't get the correct answer using it.

Answer 12

It can be tricky if there are three factors, you have to be very careful what you're calling the first factor and what you're calling the second and keep them straight. You then have to use the product rule within the product rule and it can make things get cluttered and hard to follow.

Answer 13

It's certainly possible if you pay attention and write neatly, but if another method seems like less of a headache, then use the other method.

Answer 14

I see. Well maybe I did it wrong then, I don't know. What I did was I took the first two factors and applied the product rule to them. Once I got my product, I then took that answer and multiplied it by the third factor, once again using the product rule to differentiate those two.

Answer 15

But I get how to answer the question correctly now though, so thanks.

Answer 16

Yeah, your mistake was multiplying the derivative of the first two factors by the third factor without taking into account the addition of the first two factors multiplied by the derivative of the third factor.
Wow, that sounds as complicated as it actually is.
Here's a schematic:
\[\frac{d}{dx} (u \cdot v \cdot w) = \frac{d}{dx} ((u \cdot v) \cdot w)\]
\[\rightarrow (u \cdot v) \cdot w' + w \cdot \frac{d}{dx} (u \cdot v)\]
\[\rightarrow (u \cdot v) \cdot w' + w \cdot [u \cdot v' + v \cdot u']\]

Answer 17

\[\rightarrow u \cdot v \cdot w' + u \cdot v' \cdot w + u' \cdot v \cdot w]\]

See the pattern?

Answer 18

Ah, that makes a lot of sense. But yeah, that is definitely more complicated than the method you used to solve it.

Answer 19

I agree, it's too much to remember. As easy as that pattern is to see, I'd never bother remembering it. I'd have to rederive it every time. I'd rather not waste time trying to figure out how to do it the fancy way and just break it down old school ya' know?

Answer 20

Cool, sounds good. Well thanks for the help, Cliff.