Differentiating a function inside a function. Let's say that there's a function $f(x) = g(x^2 + 5x)$, and $f'(0) = 10$. What would $g'(0)$ be here?
Because this is a function composition I tried using the chain rule in this way -
(I took the inside function $g(x)$ as u)
\begin{align}
& \ \ \ \ \ \ \ f'(x) = \frac{dg}{du}.\frac{du}{dx}\\[1em]
& \Rightarrow f'(x) = g'(x^2+5x).(2x + 5)\\[1em]
& \Rightarrow f'(0) = g'(0).(2*0 + 5)\\[1em]
& \Rightarrow 10 = g'(0).(2*0 + 5)\\[1em]
& \Rightarrow g'(0) = 2
\end{align}
 A: To preface, derivatives are not quotients.
Given the limit definition of a derivative (where $f$ is some differentiable function for all numbers in the domain of $x$) (and also $\Delta x$ is some small change in $x$)
$$\frac {df}{dx} = f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} = \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x} $$
This "limit of a quotient" cannot be a "quotient of the limits" because both numerator and denominator go to zero, so the derivative is not a quotient, but it gives intuition to go down this path anyway. There's also nonstandard analysis and all that, but it is kind of irrelevant.

If $f$ is defined as the composition of function $h$ meeting the same criteria as $f$ (a composition seen as $f(h(x))$)
Also given the definition of the chain rule with such a function $f$
$$ \frac {df}{dx} = \frac {df}{dh} \cdot \frac {dh}{dx} $$
We can work with these definitions to see how the derivatives might be treated as quotients
$$ \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x} = \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta h} \cdot \frac{\Delta h}{\Delta x} $$
$$ \hspace{3cm} =  \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta h} \cdot \lim_{\Delta x \to 0} \frac{\Delta h}{\Delta x} $$
$$ \hspace{4.4cm} =  \frac{\lim_{\Delta x \to 0} {\Delta f}} {\lim_{\Delta x \to 0} {\Delta h}} \cdot \frac{\lim_{\Delta x \to 0} {\Delta h}}{\lim_{\Delta x \to 0} {\Delta x}} $$
$$ \hspace{2cm} = \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta h} \cdot \frac{\Delta h}{\Delta x} $$
