f(x) = sqrt(x)

find f'(x)

using the definition of the derivative

To find \( f'(x) \) for \( f(x) = \sqrt{x} \) using the definition of the derivative, we proceed as follows:

### Step-by-Step Solution

The definition of the derivative is given by:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

1. **Substitute \( f(x) = \sqrt{x} \):**
   \[
   f'(x) = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h}
   \]

2. **Rationalize the numerator:** Multiply both the numerator and the denominator by the conjugate of the numerator, \( \sqrt{x+h} + \sqrt{x} \):
   \[
   f'(x) = \lim_{h \to 0} \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}
   \]

3. **Simplify the expression:** Using the difference of squares, \( (a - b)(a + b) = a^2 - b^2 \), we get:
   \[
   f'(x) = \lim_{h \to 0} \frac{(x+h) - x}{h(\sqrt{x+h} + \sqrt{x})}
   \]
   Simplify further:
   \[
   f'(x) = \lim_{h \to 0} \frac{h}{h(\sqrt{x+h} + \sqrt{x})}
   \]

4. **Cancel \( h \) in the numerator and denominator:**
   \[
   f'(x) = \lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}}
   \]

5. **Take the limit as \( h \to 0 \):** Substitute \( h = 0 \):
   \[
   f'(x) = \frac{1}{\sqrt{x} + \sqrt{x}} = \frac{1}{2\sqrt{x}}
   \]

### Final Answer
\[
f'(x) = \frac{1}{2\sqrt{x}}
\]

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions to explore this topic further:

1. How would you find the derivative of \( f(x) = x^{1/3} \) using the definition of the derivative?
2. Can you verify \( f'(x) = \frac{1}{2\sqrt{x}} \) by using the power rule instead?
3. How can you use the chain rule to find the derivative of a composite function involving square roots, like \( f(x) = \sqrt{g(x)} \)?
4. What would be the derivative of \( f(x) = \sqrt{x+2} \) using the definition of the derivative?
5. How does the derivative of \( f(x) = \sqrt{x} \) behave as \( x \) approaches zero?

**Tip:** Rationalizing the numerator is a useful technique when dealing with square roots in limits, especially for finding derivatives of root functions.

find f'(x) for f(x) = sqrt(x) using the definition of the derivative

Learn how to find the derivative of f(x) = sqrt(x) using the definition of the derivative. This step-by-step solution involves limits, rationalizing expressions, and applying the difference of squares formula. Ideal for high school calculus students.

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