The Definition of the Derivative — Question 7

Question 7

Let $$f(x) = \frac{1}{x}$$ Use the definition of the derivative to compute $f'(x)$. Show all steps clearly without using derivative rules.

Question 7 - Solution

We are given: $$f(x) = \frac{1}{x}$$

Use the definition: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h}$$

Combine the terms in the numerator: $$= \lim_{h \to 0} \frac{\frac{x - (x + h)}{x(x + h)}}{h} = \lim_{h \to 0} \frac{\frac{-h}{x(x + h)}}{h}$$

Simplify: $$= \lim_{h \to 0} \frac{-1}{x(x + h)}$$

Now take the limit as $h \to 0$: $$f'(x) = \frac{-1}{x^2}$$

Final Answer: $$f'(x) = \boxed{-\frac{1}{x^2}}$$