The Definition of the Derivative — Question 8

Question 8

Let $$f(x) = \sqrt{x}$$ Use the definition of the derivative to compute $f'(x)$. Do not use derivative shortcuts. Show all algebraic simplification and steps clearly.

Question 8 - Solution

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

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

Multiply numerator and denominator by the conjugate: $$= \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}} = \lim_{h \to 0} \frac{(x + h) - x}{h(\sqrt{x + h} + \sqrt{x})}$$

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

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

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