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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in order to fully understand the proof of Theorem 8.15 ...

Theorem 8.15 and its proof read as follows:

View attachment 9416

View attachment 9417In the above proof by Browder we read the following:" ... ... Now

\(\displaystyle (g \circ f) (p + h) - (g \circ h) (p) = g(q + k) - g(q)\) ... ... "

Can someone please show how/why \(\displaystyle (g \circ f) (p + h) - (g \circ h) (p) = g(q + k) - g(q)\) ... ...

Help will be appreciated ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in order to fully understand the proof of Theorem 8.15 ...

Theorem 8.15 and its proof read as follows:

View attachment 9416

View attachment 9417In the above proof by Browder we read the following:" ... ... Now

\(\displaystyle (g \circ f) (p + h) - (g \circ h) (p) = g(q + k) - g(q)\) ... ... "

Can someone please show how/why \(\displaystyle (g \circ f) (p + h) - (g \circ h) (p) = g(q + k) - g(q)\) ... ...

Help will be appreciated ...

Peter