Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $X$? for example, if $\phi(X)=stand(X)=\frac{X-E(X)}{std(X)}$ (suppose $E(X)$ and $std(X)$ exist), how to calculate $\frac{d\phi(X)}{dX}$? I knew that if we assume the approx. distribution of $X$ exists using LLN or CLT, we can just using mean and std of Gaussian distribution. But how to get a general form of the derivative $d(stand(X))/dX$, $d(EX)/X$ and $d(std(X))/dX$? how to define the limitation and rules of derivatives in this case?
Background of this problem:
Suppose I trained a neural network with standardisation of the data following $(X-EX)/std(X)$. The input is $x(t)$ and output is $y(t)$. How can I calculate the sensitivity of this trained network (basically the $dy(t)/dx(t)$)? if I multiply $(1+\epsilon)$ to $x(t)$, the $1+\epsilon$ will disappear after standardisation. So what should I do? I tried to add a perturbation to the standardized input $X'$ which is $(X-mean(X))/(std(X))$. But after the perturbation what I got is $dy(t)/d(standard(x(t)))$ rather than $dy/dx$. So I also need the ratio of $dx/d(standard(x))$ where $standard(x)$ is $(x(t) - mean(x(t)))/std(x(t))$.
We can assume $x(t)$ and $y(t)$ is a $N$-order Markov process.
So how to calculate the $dx/d(standard(x))$? thanks a lot!
