Derivative of random normal times indicator function

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 $frac{dE[f(X)]}{dX} = E[Xdelta_{X} + 1_{X>a}]$ where $delta_{X}$ is the delta function. Is this in the right direction? Any hint is appreciated.