We all know the stock-standard and conventional differentiation rules, such as the Sum and Difference Rule, Product Rule, Chain Rule etc. But are there other more advanced rules that are not treated in most introductory Calculus textbooks?
For example lets, say I have two arbitrary single-variable functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$, how would one find the derivatives of the following functions, and furthermore are there rules for differentiating functions of these sort of nature? $$\text{Straightforward Examples}$$ $$f(x) = \sum_{i=0}^{n}\ g_{i}(x)$$ $$f(x) = \prod_{i=0}^{n}\ g_{i}(x)$$ $$f(x) = {g(x) \choose x} $$
And some more complicated examples, here, $h: \mathbb{R} \to \mathbb{R}$ $$\text{More Complicated Examples}$$ $$f(x) = g(x)^{\sum_{i=0}^{n}\ h_{i}(x) }$$ $$f(x) = g(x)^{\prod_{i=0}^{n}\ h_{i}(x)}$$ $$f(x) = {\sum_{i=0}^{n}\ g_{i}(x) \choose x} $$ $$f(x) = \prod_{i=0}^{n }\ \cdot \prod_{i=0}^{m}\ g_{i}(x)$$ $$f(x) = \sum_{i=0}^{n}\ g_{i}(x) \cdot {\prod_{i=0}^{m}\ g_{i}(x) \choose x}$$
Are there extensions of the usual differentiation rules, to find the derivatives for these kinds of functions? If not how would one use the usual differentiation rules to find the derivatives of these kinds of functions?
I spent some time working on analogs of integration by parts that can be created using the quotient rule instead of the product rule, and versions based on more general versions of the product rule, but found that they were functionally no more or less useful than the well-known version of integration by parts.
The straightforward examples you gave, (first two at least) are easy to do by repeated use of the familiar sum and product rules for differentiation. The third expression does not make sense unless $x$ is an integer, so I doubt you can naturally extend from that to a function that is differentiable.
For the more complicated expressions:
The first can be handle by use of logarithmic differentiation (to handle variable both in and outside of exponent) and the chain rule and sum rule.
Likewise the second, but using the product rule instead of sum rule.
The third does not make sense unless $x$ is an integer, so you probably don't have an easy extension to a differentiable function in any natural way there.
The fourth is just repeated application of the product rule (helps to have done the straightforward version already).
The last again doesn't make sense unless $x$ is an integer.