Let $X,Z$ be random variables with probability density functions $p_X,p_Z$. Suppose $Z=f(X)$, where $f$ is continuous and differentiable. How is $p_Z$ related to $p_X$? It's tempting to say $p_Z(z) = p_X(f^{-1}(z))$, but I think that is not correct: I think it might be
$$p_Z(z) = {p_X(f^{-1}(z)) \over f'(f^{-1}(z))},$$
where $f'$ is the derivative of $f$, but I am not sure whether I've got that right. What is the correct rule?