Mathematically, a function take in exactly 2 input, and they have no meanings whatsoever. So x' is just another parameter, just like x. Partial derivative of f with respect to x is indeed x', and there are no confusion, because x' is just a name for a parameter with no relationship to x.

In the context of physics and engineering, people usually write function over other variables with relation to each other, like this. This is when the confusion happen; if only they are written up to math's standard of clarity the confusion could have been avoided. The rule of thumb for interpretation is, for partial derivative, take the derivative as written (ignoring the relationship with each other); for total derivative, you have to expand out the relationship. So in this case, you still have partial derivative of f with respect to x being x'.

Another way to say it is: partial derivative is about the property of a function, the relationship between its output and its input. Physics and engineering mix together the definition of a function with the purpose of the function, which is what they are going to use the function on. But the purpose have no effects on the definition, and hence no effects on partial derivative. But when you take total derivative with respect to time, you're expected to understand that you are actually constructing a new function by applying the old function for its intended purpose.
**I think you are confusing yourself by calling the function "x" for x, instead of something else (like g).

What you then have, is g, and its derivative g', both functions of time.

You now define a third function f = gg'. Importantly, f is still just a function of time, since f(t) = g(t)g'(t) (if not, then I'm misunderstanding your question).

You can then calculate the derivative of f (w.r.t. time) using the product rule.
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