Proving a theorem: Changing the Order of Differentiation (large question)

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.

Next, go on to explain how to obtain the following expression:

where as .

Explain the derivation of the similar expression

where as ,

using the fact that is differentiable at .

With all this, complete the proof of the theorem.

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This is a lot of info to work with, I know. But I can't summarize it any better than this because the question has such a high context requirement.