Function Plus a Constant as a Parameter of a Function We spent about 10 minutes arguing this in Calculus class, but we ended up dismissing the problem. Here it is:
We were trying to prove the chain rule from 1st principles, but we weren't sure which equation was right:
$h'(x)=\lim\limits_{x\to 0}\frac{f(g(x) + h) - f(g(x))}{h}$ or $h'(x)=\lim\limits_{x\to 0}\frac{f(g(x + h)) - f(g(x))}{h}$
Basically, would the $+h$ be included or excluded in $g(x)$?
Note: the original equation is: $f'(x)=\lim\limits_{x\to 0}\frac{f(x + h) - f(x)}{h}$

It appears your $h$ function is $h(x) = f(g(x))$. If so, then note that whatever you replace $x$ with on one side must be the same on the other side, e.g., $h(y) = f(g(y))$, $h(x + j) = f(g(x + j))$, etc. Thus, the correct way to express its derivative is
$$\begin{equation}\begin{aligned}
h'(x) & = \lim_{j \to 0}\frac{h(x + j) - h(x)}{j} \\
& = \lim_{j \to 0}\frac{f(g(x+j)) - f(g(x))}{j}
\end{aligned}\end{equation}\tag{1}\label{eq1A}$$
Also note that you are using $x \to 0$ in your limits instead of the correct $h \to 0$. In addition, I used $j$ instead of $h$ as the limiting value to avoid confusion with the $h(x)$ function.
