To find the derivative of $f(x) = \frac{1}{\sqrt{x}}$ using the definition of the derivative, we start with the standard formula for the derivative:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Step-by-Step Process:

1. Set up the function: The given function is $f(x) = \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$.

2. Substitute into the definition: Using the definition of the derivative:

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

3. Combine the two terms in the numerator: To combine the terms, we need to get a common denominator in the numerator:

   $\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}} = \frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x} \cdot \sqrt{x+h}}$

   So, the expression becomes:

   $f'(x) = \lim_{h \to 0} \frac{\frac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x} \cdot \sqrt{x+h}}}{h}$

   Simplifying:

   $f'(x) = \lim_{h \to 0} \frac{\sqrt{x} - \sqrt{x+h}}{h \cdot \sqrt{x} \cdot \sqrt{x+h}}$

4. Multiply by the conjugate: To simplify $\sqrt{x} - \sqrt{x+h}$, multiply the numerator and denominator by the conjugate $\sqrt{x} + \sqrt{x+h}$:

   $f'(x) = \lim_{h \to 0} \frac{(\sqrt{x} - \sqrt{x+h})(\sqrt{x} + \sqrt{x+h})}{h \cdot \sqrt{x} \cdot \sqrt{x+h} \cdot (\sqrt{x} + \sqrt{x+h})}$

   Using the difference of squares formula in the numerator:

   $(\sqrt{x})^2 - (\sqrt{x+h})^2 = x - (x+h) = -h$

   So, the expression simplifies to:

   $f'(x) = \lim_{h \to 0} \frac{-h}{h \cdot \sqrt{x} \cdot \sqrt{x+h} \cdot (\sqrt{x} + \sqrt{x+h})}$

5. Cancel the $h$ terms: The $h$ in the numerator and denominator cancels:

   $f'(x) = \lim_{h \to 0} \frac{-1}{\sqrt{x} \cdot \sqrt{x+h} \cdot (\sqrt{x} + \sqrt{x+h})}$

6. Evaluate the limit as $h \to 0$: As $h \to 0$, $\sqrt{x+h} \to \sqrt{x}$, so the expression becomes:

   $f'(x) = \frac{-1}{\sqrt{x} \cdot \sqrt{x} \cdot (2\sqrt{x})}$

   Simplifying:

   $f'(x) = \frac{-1}{2x^{3/2}}$

Final Answer:

$f'(x) = -\frac{1}{2x^{3/2}}$

This is the derivative of $f(x) = \frac{1}{\sqrt{x}}$ using the definition of the derivative.

Would you like further explanation or more details?

Related Questions:

1. How would you find the derivative of $\frac{1}{x^2}$ using the definition of the derivative?
2. What is the interpretation of a derivative in terms of rates of change?
3. How can you use the power rule to verify the derivative of $x^{-\frac{1}{2}}$?
4. How does multiplying by a conjugate help simplify radical expressions?
5. Can you apply the definition of the derivative to a trigonometric function like $\sin(x)$?

Tip:

Always simplify complex fractions by multiplying by the conjugate when dealing with square roots—it helps eliminate the radicals in the numerator.