I have to find $\frac{dE[f(X)]}{dX}$ where $f(X) = X1_{X>a}$ where $X \sim N(0,1)$ , $1_{X>a}$ is an indicator function taking value 1 if $X>a$ and $0$ otherwise, and $a$ is some constant. I have trouble understanding how to differentiate a random variable (standard normal in this case). On the top of it, an indicator function of random variable is involved.
My approach: Use simple chain rule first and get $\frac{dE[f(X)]}{dX} = E[X\delta_{X} + 1_{X>a}]$ where $\delta_{X}$ is the delta function. Is this in the right direction? Any hint is appreciated.
$E(f(X))=\int_a^{\infty} x \phi (x)dx$ where $\phi$ is the standard normal density. The derivative of this w.r.t. $a$ is $-a\phi(a)=-\frac a {\sqrt 2\pi} e^{-a^{2}/2}$.