Suppose we have two functions with a symmetrical behaviour with respect to an axis. For the sake of simplicity, let $f(x)$ and $g(x)$ have a symmetrical behaviour with respect to the $y$ axis.
A trivial example could be $f(x) = e^x$ and $g(x) = e^{-x}$.
1) Is there a way to obtain a general relation between their derivates?
In other words, we have
$$\displaystyle \frac{df(x)}{dx} = h(x) = e^x$$ $$\displaystyle \frac{dg(x)}{dx} = i(x) = -e^{-x}$$
is there a way to write a general relation that in such cases $h(x)$ and $i(x)$ always satisfy? I thought that was $h(x) = -i(x)$ because of the opposite behaviour of $f(x)$ and $g(x)$ with respect to $x$, but it is wrong.
2) If a relation does exist, is it possible to extend it to a multiple-variable case?
Say, $f(x,y,z)$ and $g(x,y,z)$ have a symmetrical behaviour with respect to the plane $z = 0$. If we evaluate their partial derivatives
$$\displaystyle \frac{\partial f(x,y,z)}{\partial z} = h(x,y,z)$$ $$\displaystyle \frac{\partial g(x,y,z)}{\partial z} = i(x,y,z)$$
can we write a relation between $h(x,y,z)$ and $i(x,y,z)$?
Well you essentially have $g(x)=f(-x)$ and hence by the chain rule, $g'(x)=-f'(-x)$. I suspect this is enough for you to be able to determine any extensions, etc. yourself.