Derivative of principal value distribution $1/x$ is equal to finite part distribution $-1/x^2$?
Your computation is correct. Someone dropped the minus sign when stating the exercise. Remembering the calculus formula $(1/x)=-1/x^2$ would help prevent that. The calculus formula doesn't prove the result for distributions, but it must be consistent with it in the sense that differentiation away from singularities is the same as classical derivative.