What is the density function of $V=U^{-a}$, where $a>0$ and $U$ is a uniform random variable?

Question

I'm working through some of the questions in the textbook, "Mathematical Statistics and Data Analysis" by John Rice and am having some difficulties with this question.

I start off with $F_V(v)=P(V\leq v)=P(U^{-a}\leq v)=P(U\geq v^{-1/a})=1-P(U<v^{-1/a})=F_U(v^{-1/a})$

Assuming what I did above is correct, should I be taking the derivative of $F_U(v^{-1/a})$ to find $f_v(v)$?

Am I done after taking the derivative or do I need to find the domain of the variable? How would you find the domain?