I’m self-studying Stewart’s calculus and I went back to review the proof of the chain rule, but I keep getting confused about a particular line.
The proof starts as follows:
$\Delta y = f(x+\Delta x) - f(x)$: this part is clear to me.
$\displaystyle{\lim_{\Delta x \to 0}} [\Delta y/\Delta x] = f’(a)$: this part is clear to me.
let $\varepsilon$ be a number such that $\varepsilon = (\Delta y/\Delta x) - f’(a)$: not exactly sure why they decided to go in this direction. How does one go about logically arriving at this step?
$\Delta y = \varepsilon \Delta x + \varepsilon f’(a)$: this is clear to me.
If we define $\varepsilon$ to be 0 when $\Delta x = 0$ then $\varepsilon$ becomes a continuous function of $\Delta x$: I’m also confused here. How can we define $\varepsilon$ to be 0 when $\Delta x = 0$? Not sure what that means. Further, if $\Delta x = 0$ then wouldn’t $\varepsilon = 0$ (no change in the difference quotient) $- f’(a) = -f’(a)$?
#3 and #5 are preventing me from proceeding. Can anyone help out?