Derivative in the sense of distribution of an integral w.r.t. a measure

Question

I want to take the derivative w.r.t. $x$ of the following function in the sense of distribution: $C(t,x)=\int(y-x)_+ d\mu_t(y)$. The goal is to recover the distribution function of $\mu_t$, namely $\mu_t((-\infty,x])=...$. Can someone help me in this point? I am not so familiar with derivatives in the sense of distributions...