I'm reading Vector Calculus from http://mecmath.net/. This is a free PDF book for students of Calculus III. In section 2.4 (page 78) it introduces the directional derivative and theorem 2.2:
So far so good and I get the intuition. I am having trouble following the proof of the theorem though. Please see following picture:
I understand the first equality; a partial w.r.t. $y$ is being taken so $x$ is being held constant at $a + hv_1$ therefore it can be considered the derivative of a function $g(y) = f(a+hv_1,y)$.
I'm confused where $\alpha$ goes after the 2nd equals sign. I'm confused that $g\prime$ is claimed to be equal to some difference quotient and not the limit of the difference quotient. If anyone could talk me through what's going on here I'd be grateful.