Reciprocal Rule Derivative using First Principle of Derivatives

The Mathematician — Sat, 06 May 2023 13:00:32 +0000

How to prove reciprocal rule derivative using first principle of derivatives?

We will prove the reciprocal rule by the first principle method, the definition of the derivative. The Reciprocal Rule is defined as

\begin{equation*} \bigg(\frac{1}{f}\bigg)' (x) = \frac{-f'(x)}{(f(x))^2}. \end{equation*}

We will prove that this equality holds.

\begin{align*} \bigg(\frac{1}{f}\bigg)' (x) &= \lim_{h \rightarrow 0} \frac{\frac{1}{f(x + h)} - \frac{1}{f(x)}}{h} \\ &= \lim_{h \rightarrow 0} \frac{\frac{f(x) - f(x + h)}{f(x + h)f(x)}}{h} \\ &= \lim_{h \rightarrow 0} \frac{(- f(x+h) + f(x))}{h} \cdot \lim_{h \rightarrow 0} \frac{1}{f(x + h)f(x)} \\ &= - \lim_{h \rightarrow 0} \frac{(f(x+h) - f(x))}{h} \cdot \lim_{h \rightarrow 0} \frac{1}{f(x + h)f(x)} \\ &= f'(x) \cdot \lim_{h \rightarrow 0} \frac{1}{f(x + h)f(x)} \\ &= f'(x) \cdot \frac{1}{f(x)f(x)} \\ &= \frac{-f'(x)}{(f(x))^2}. \end{align*}

This shows that the Reciprocal Rule indeed holds, which completes the proof.