I am reading a proof of the Chain Rule in my text and I see this:
I get the first three lines, but then I have an issue here:
Why does $\epsilon$ (the difference between the difference quotient and the slope at a imply the next equation? Why does $\Delta{y} = f'(a)\Delta{x} + e\Delta{x}$. Isn't the (change in y) just the slope * (change in x)? Why do we need the $e\Delta{x}$
On a high level, what is going on there? Is it because the slope at a * x only gives you part of the change in y? Is $f'(a)\Delta{x}$ an incomplete picture of the change in y? Why?
I also don't understand the significance of this line:
If we define $\epsilon$ to be 0 when $\Delta x$ = 0, then $\epsilon$ becomes a continuous function of Dx.