Find n so that the equation V=r^n (3cos²theta-1) satisfies the equation?

n = 2, -3. Easier than it looks.
How to get there shall be your homework.

Just kidding. Couple hints (might err on the terms, haven't studied math in english):
The by r / by T derivative means that you can treat the other variable as a constant, in this case quite simplifying things with these basic functions.
Use trig identities, saves grunt work which is nevertheless possible using derivative of multiplied functions.

Initially thought that was a dead giveaway for sin2(t) + cos2(t) = 1 identity; wishful thinking on another examination, results in much more grunt work [solves to 1/2 * (-sqr(sin2(T)-12cos(2T)-4) / sin(T) - 1) eventually].
Consider that an extra tip - these tiny lapses in attention are the bane of straightforward calculations, principally in exams.