This question is right out of Taylor & Mann Advanced Calculus, Third Edition, and is a REALLY tricky one. Due to how context-heavy this is, I can't write it word-for-word.

We're to prove the following theorem:

Let

and its first partial derivatives

be defined in a neighborhood of the point

, and suppose that

and

are differentiable at that point. Then

.

This theorem uses part of another theorem's proof to start things off, before getting to the parts that I can't figure out.

Let

be a number different from zero such that the point

is inside a square having its center at

. We then consider the following expression:

If we introduce the function

,

we can express

in the form

(*)

Now

has the derivative

Hence

is continuous, and we may apply the mean-value theorem for derivatives to (*), obtaining the following:

, where

That ends the part of the separate proof; now to parts which I'm in the dark about (these are from the actual question).

From the fact that

is differentiable at

, one can write

, where

as

.

Explain why this is so.