I'm wondering about the form that a limits definition of partial derivatives should take when we partially derivate a function of dependent and independent variables wrt to an independent variable.
More precisely stated, the question is: Given two functions $\{f,g\}\in \mathcal{C}^1$ (such that their first derivatives exist, and such that $f$ has partial derivatives everywhere) let: $$f'=\frac{\partial}{\partial x}f(x,g(x))$$ Expanding the RHS we get: $$f'=\frac{\partial f}{\partial g}\frac{\partial g}{\partial x}+\frac{\partial f}{\partial x}$$ Both $\frac{\partial f}{\partial g}$ and $\frac{\partial g}{\partial x}$ have a precise definition using limits, so the first term its not problematic... But i do not know how to define the second term in terms of limits. Should i use: $$\frac{\partial f}{\partial x}=\lim_{∆\rightarrow0}\frac{f(x+∆,g(x+∆))-f(x,g(x))}{∆}$$ or rather: $$\frac{\partial f}{\partial x}=\lim_{∆\rightarrow0}\frac{f(x+∆,g(x))-f(x,g(x))}{∆}$$ Wich of the previous identities gives the right equation to $f'$?