Product and Quotient Rule proof using linearisation

Question

So I've recently been introduced to the concept of linearization and now I'm beginning to apply this concept to prove certain differenation rules. I've managed to prove the chain rule so far, but I really can't think how to prove the product or quotient rule using linearization. Does anyone have any hints as to how to go about proving these rules? Will the proof of the chain rule come in handy?

Answer 1

Product rule:

Let $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + O(|x - x_0|^2) $$ and $$ g(x) = g(x_0) + g'(x_0)(x - x_0) + O(|x - x_0|^2). $$ Then, \begin{multline} f(x)g(x) = \left[f(x_0) + f'(x_0)(x - x_0) + O(|x - x_0|^2)\right] \cdot \\ \left[g(x_0) + g'(x_0)(x - x_0) + O(|x - x_0|^2)\right] \end{multline}

So, $$ f(x)g(x) = f(x_0) g(x_0) + \color{blue}{[f(x_0) g'(x_0) + f'(x_0) g(x_0)}](x - x_0) + O(|x - x_0|^2). $$

Quotient rule can be proved as corollary, or you can do $$ \frac{f(x)}{g(x)} = a_0 + a_1 (x - x_0) + O(|x - x_0|^2), $$ expand both $f$ and $g$, and determine the constants by equating the powers.