Quiz 3 - Definition of a derivative

Problem 1

Question

Use the definition of a derivative to find the derivative \(f'(x)\) of the function \(f(x) = \frac{1}{x}\).

Solution

Using the definition: \[ \begin{align*} f'(x) &= \lim_{h\rightarrow 0}\left(\frac{f(x + h) - f(x)}{h}\right) \\ &= \lim_{h\rightarrow 0}\left(\frac{\frac{1}{x+h} - \frac{1}{x}}{h}\right) \\ &= \lim_{h\rightarrow 0}\left(\frac{\frac{x}{x(x+h)} - \frac{x+h}{x(x+h)}}{h}\right) \\ &= \lim_{h\rightarrow 0}\left(\frac{\frac{x - (x+h)}{x(x+h)}}{h}\right) \\ &= \lim_{h\rightarrow 0}\left(\frac{\frac{ -h}{x(x+h)}}{h}\right) \\ &= \lim_{h\rightarrow 0}\left(\frac{-1}{x(x+h)}\right) \\ \Aboxed{f'(x) &= -\frac{1}{x^2}} \end{align*} \]