Coulomb's Law and spheres Problem

[frankfjf]

Homework Statement

Of the charge Q initially on a tiny sphere, a portion q is to be transferred to a second, nearby sphere. Both spheres can be treated as particles. For what value of q/Q will the electrostatic force between the two spheres be maximized?

Homework Equations

(k)(q1q2)/r^2

The Attempt at a Solution

I posted a similar problem concerning an electric field some time ago, and am attempting to use the same solution, that is, taking the derivative of the equation I end up with, but the derivative ends up being zero for me. I end up with:

k(Q-q)q / r^2

And then after factoring out k/r^2, I'm left with just (Q-q)q. I tried doing a derivation using the product rule, but I just end up with zero. When I try to derive this, I get:

1 * (1-1) + (Q-q) which just leaves me with Q - q, which is not the answer. What am I doing wrong?

 

[neutrino]

You differentiated the equation with respect to...?

 

[frankfjf]

Ah nevermind, upon multiplying the q back into the (Q-q) and then deriving, I get the correct answer. However, can it be said as a rule of thumb that you should factor AFTER taking the derivative, or are there situations where you should factor beforehand?

 

[frankfjf]

You differentiated the equation with respect to...?

Basically I used df/dq.

 

[neutrino]

However, can it be said as a rule of thumb that you should factor AFTER taking the derivative, or are there situations where you should factor beforehand?

There's no difference between: $$\frac{d}{dq}\left(q(Q-q)\right)$$ and $$\frac{d}{dq}\left(qQ-q^2\right)$$, and this holds true no matter what kind of functions are in the product. (They should be continuous of course.)