Factoring Difficulties: Derivative of $(3x+1)^2(2x-3)^3$

[DeusAbscondus]

I'm trying to find the derivative of:

$$f(x)=(3x+1)^2(2x-3)^3 \text{ by using the product method}$$

Here is my working out so far, using product rule:$u'v+uv'$
$$\frac{d}{dx} (3x+1)^2(2x-3)^3= 2(3x+1)\cdot 3 *(2x-3)^3+(3x+1)^2\cdot3(2x-3)^2\cdot2$$
Simplified: $$f'(x)=6(3x+1)(2x-3)^3+6(3x+1)^2(2x-3)^2$$

At this point, I have a question:
1. is it best practice to factor out $(2x-3)^3$ or $(2x-3)^2$?
Is there a general answer to this question, valid for all such factoring situations?

To continue with my calculations, I will factor out the lower exponential factor:

$$\Rightarrow 6(3x+1)(2x-3)^2[(2x-3)+(3x+1)]$$
$$\Rightarrow (18x+6)(2x-3)^2(5x-2)$$

Final question: is there any obvious problem with (or improvement to be made) in the way I have set this out? something I'm doing which could be avoided/changed so as to avoid careless errors creeping in?
I ask, because this took me an inordinate amount of time to get right.
(It is correct, is it not, by the way?)

thanks,
DeusAbs

 

[CaptainBlack]

I'm trying to find the derivative of:

$$f(x)=(3x+1)^2(2x-3)^3 \text{ by using the product method}$$

Here is my working out so far, using product rule:$u'v+uv'$
$$\frac{d}{dx} (3x+1)^2(2x-3)^3= 2(3x+1)\cdot 3 *(2x-3)^3+(3x+1)^2\cdot3(2x-3)^2\cdot2$$
Simplified: $$f'(x)=6(3x+1)(2x-3)^3+6(3x+1)^2(2x-3)^2$$

At this point, I have a question:
1. is it best practice to factor out $(2x-3)^3$ or $(2x-3)^2$?
Is there a general answer to this question, valid for all such factoring situations?

You take out the greatest common factor of the terms, which is: \(6(3x+1)(2x+3)^2\) to give:

\[\begin{aligned}6(3x+1)(2x-3)^3+6(3x+1)^2(2x-3)^2&=6(3x+1)(2x-3)^2[(2x-3)+(3x+1)]\\ &=6(3x+1)(2x-3)^2(5x-2)\end{aligned}\]

CB

 

[DeusAbscondus]

You take out the greatest common factor of the terms, which is: \(6(3x+1)(2x+3)^2\)
CB

This is something I learned (greatest common factor) on my own out of a textbook (ie: outside class situation and in a class where no reference has been made to GCF) and now have to relearn by trial and error because of:

1. faulty memory
2. lack of principle-guided (axiomatic) coaching at present

Thanks for helping to supply this deficiency Cap'n.

DeusAbs
(Back to the grind, pleasantly, after 3 weeks of no sums!)