Differentiate a function of random variables Suppose I have some function $V(x)=x+log(c)$, where $x$ is a continuous random variable and $c$ a constant bounded on $[0,1]$. I have some queries regarding the following:
i) May the above function be differentiated in the conventional manner w.r.t $x$ even though the resultant derivative will be random?
ii)  How may I determine if $V(x)$ is differentiable?
iii) What is the derivative of $V(x)$?
Note: $x$ is bounded on $[a,b]$
Any help would be greatly appreciated.   Thank You
 A: Let $X$ be a random variable on probability space $(\Omega,\mathcal A,P)$.
Then $X$ is a measurable function $\Omega\to\mathbb R$, so it takes values in $\mathbb R$.
If $V$ denotes a suitable function $\mathbb R\to\mathbb R$ then $V(X)=V\circ X$ can be looked at as another random variable prescribed by: $$\omega\mapsto V(X(\omega))$$
It is quite well possible that the function $V:\mathbb R\to\mathbb R$ is differentiable.
If $V'$ denotes its derivative then in $V'(X)=V'\circ X$ again we find a random variable. 
This time prescribed by:$$\omega\mapsto V'(X(\omega))$$
In your case we have $V$ prescribed by $x\mapsto x+\log c$ so $V'$ is prescribed by $x\mapsto1$.
So in your situation $V'\circ X$ is the (degenerated) random variable prescribed by $$\omega\mapsto 1$$
On the other hand $V(X)=V\circ X$ is a function $\Omega\to\mathbb R$ and normally outcome space $\Omega$ is not equipped with structure that allows differentiation.
Be aware of the distinction between $V$ (a function $\mathbb R\to\mathbb R$) and $V\circ X$ (a function $\mathbb \Omega\to\mathbb R$).
A: $$\frac{dV(x)}{dx}=1$$
Randomness of $x$ does not matter. (And in this particular case, even the value of $x$ does not matter.)
