Derive the formulas for the derivative of a product and the derivative of a quotient from the corresponding formulas for the derivative of a sum and the derivative of a difference.
We know the derivative rules for sums and differences are:
![\[ \left( f(x) + g(x) \right)' = f'(x) + g'(x) \quad \text{and} \quad \left( f(x) - g(x) \right)' = f'(x) - g'(x). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-564c4d6f1b89baee6710b9deb9137f19_l3.png "Rendered by QuickLaTeX.com")
To derive the derivative rule for products using logarithmic differentiation we let 
and compute
![\begin{align*} h(x) = f(x)g(x) && \implies && \log |h(x)| &= \log |f(x)g(x)| \\[9pt] && \implies && (\log |h(x)|)' &= (\log |f(x)g(x)|)' \\[9pt] && \implies && \frac{h'(x)}{h(x)} &= (\log |f(x)| + \log |g(x)|)' \\[9pt] && \implies && \frac{h'(x)}{h(x)} &= \frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} \\[9pt] && \implies && h'(x) &= \left( \frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} \right) h(x) \\[9pt] && \implies && h'(x) &= \left( \frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} \right) f(x)g(x) \\[9pt] && \implies && (f(x)g(x))' &= f'(x) g(x) + f(x)g'(x). \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b9759d2886cccf3d8f920af4e334bbf6_l3.png "Rendered by QuickLaTeX.com")
This is the usual rule for derivative of a product.
Similarly, for the derivative of a quotient, let 
and then compute,
![\begin{align*} h(x) = \frac{f(x)}{g(x)} && \implies && \log | h(x)| &= \log \left| \frac{f(x)}{g(x)} \right| \\[9pt] && \implies && (\log|h(x)|)' &= \left( \log \left| \frac{f(x)}{g(x)} \right| \right)' \\[9pt] && \implies && \frac{h'(x)}{h(x)} &= \left( \log |f(x)| - \log |g(x)| \right) ' \\[9pt] && \implies && \frac{h'(x)}{h(x)} &= \frac{f'(x)}{f(x)} - \frac{g'(x)}{g(x)} \\[9pt] && \implies && h'(x) &= \left( \frac{f'(x)}{f(x)} - \frac{g'(x)}{g(x)} \right) h(x) \\[9pt] && \implies && h'(x) &= \left( \frac{f'(x)}{f(x)} - \frac{g'(x)}{g(x)} \right) \frac{f(x)}{g(x)} \\[9pt] && \implies && \left( \frac{f(x)}{g(x)} \right)' &= \frac{f'(x)g(x) - g'(x)f(x)}{f(x)g(x)} \cdot \frac{f(x)}{g(x)} \\[9pt] && \implies && \left( \frac{f(x)}{g(x)} \right)' &= \frac{f'(x)g(x) - g'(x)f(x)}{(g(x))^2}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-91fc7325d9d121b18dbc4f1135e64da6_l3.png "Rendered by QuickLaTeX.com")
Which is the usual rule for derivative of a quotient.