The Definition of the Derivative — Question 5

Question 5

Let $$f(x) = \frac{1}{x}$$

Use the definition of the derivative to find an expression for $f'(x)$. Then evaluate $f'(2)$ using your expression.

Question 5 - Solution

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

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

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

Take the limit: $$f'(x) = \frac{-1}{x^2}$$

Now evaluate at $x = 2$: $$f'(2) = \frac{-1}{2^2} = \frac{-1}{4}$$

Final Answer: $$f'(x) = \frac{-1}{x^2}, \quad f'(2) = \boxed{-\frac{1}{4}}$$