# Partial derivatives of random variables

I have a question about partial derivatives and random variables, say we have two random variables, $$r$$ and $$s$$, and they are related by the expression $$s=e^r$$. Now, $$r$$ has an explicit dependency on the variable $$t$$, $$r(t)$$. Hence, $$s$$ doesn't explicitly depend on the variable t. Say we want to compute $$\partial s/\partial t$$. Can I just use the chain rule? I have seen some arguing that this is zero, but I'm not convinced. Note that I want the partial derivative, not the total derivative. Thank you.

I think the confusion comes from how you define the partial derivative. Usually, in calculus books, you have something like $$z=f(x,y), x=g(u,v),$$ and $$y=h(u,v)$$, and you can use the chain rule to find $$\partial z/\partial u$$. When you do, the definition of the “partial” derivative is that $$v$$ is held constant. You would NOT define $$\partial z/\partial u$$ as holding $$x$$ and $$y$$ constant.
However, in physics we often have a quantity that depends on position and time, with position being a function of time, say $$f(x(t), t)$$. If for a particular function $$f$$ there was no explicit time dependence, then we say $$\partial f/\partial t =0$$; we wouldn’t use the chain rule with $$x$$ because we’d be holding $$x$$ constant.
I should add that I’m not sure how “random variables” factors into your question. $$r$$ seems to be a deterministic function of $$t$$.