use the def of derivative to find the derivative of (4x-3)^1/2

To find the derivative of \( f(x) = \sqrt{4x - 3} \) using the definition of the derivative, we follow these steps.

### Definition of the derivative:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]
For \( f(x) = \sqrt{4x - 3} \), we have:

\[
f'(x) = \lim_{h \to 0} \frac{\sqrt{4(x+h) - 3} - \sqrt{4x - 3}}{h}
\]

### Step-by-step process:

1. **Simplify the expression:**
   \[
   f'(x) = \lim_{h \to 0} \frac{\sqrt{4x + 4h - 3} - \sqrt{4x - 3}}{h}
   \]

2. **Rationalize the numerator:**
   Multiply both the numerator and denominator by the conjugate of the numerator:
   \[
   \text{Conjugate: } \sqrt{4x + 4h - 3} + \sqrt{4x - 3}
   \]
   Applying this:
   \[
   f'(x) = \lim_{h \to 0} \frac{\left(\sqrt{4x + 4h - 3} - \sqrt{4x - 3}\right) \left(\sqrt{4x + 4h - 3} + \sqrt{4x - 3}\right)}{h \left(\sqrt{4x + 4h - 3} + \sqrt{4x - 3}\right)}
   \]

3. **Simplify using the difference of squares:**
   \[
   f'(x) = \lim_{h \to 0} \frac{(4x + 4h - 3) - (4x - 3)}{h \left(\sqrt{4x + 4h - 3} + \sqrt{4x - 3}\right)}
   \]
   Simplifying the numerator:
   \[
   f'(x) = \lim_{h \to 0} \frac{4h}{h \left(\sqrt{4x + 4h - 3} + \sqrt{4x - 3}\right)}
   \]

4. **Cancel \( h \) in the numerator and denominator:**
   \[
   f'(x) = \lim_{h \to 0} \frac{4}{\sqrt{4x + 4h - 3} + \sqrt{4x - 3}}
   \]

5. **Take the limit as \( h \to 0 \):**
   As \( h \to 0 \), the term \( \sqrt{4x + 4h - 3} \to \sqrt{4x - 3} \). Therefore:
   \[
   f'(x) = \frac{4}{2\sqrt{4x - 3}}
   \]

6. **Simplify the final result:**
   \[
   f'(x) = \frac{2}{\sqrt{4x - 3}}
   \]

Thus, the derivative of \( f(x) = \sqrt{4x - 3} \) is:
\[
f'(x) = \frac{2}{\sqrt{4x - 3}}
\]

Would you like more details or clarification on any step?

Here are 5 related questions for practice:
1. Use the definition of the derivative to find \( \frac{d}{dx} (\sqrt{2x + 1}) \).
2. Compute the derivative of \( \sqrt{x^2 + 5x} \) using first principles.
3. Find the derivative of \( (x^3 - 2)^2 \) using the chain rule.
4. Calculate \( \frac{d}{dx} \ln(4x - 3) \) directly and using the definition.
5. Differentiate \( 2\sqrt{3x + 1} \) and compare it with the original problem.

**Tip**: The chain rule is a powerful tool for differentiating composite functions like \( \sqrt{4x - 3} \).

Use the definition of derivative to find the derivative of (4x-3)^1/2.

Find the derivative of √(4x - 3) using the definition of the derivative, including steps to simplify the expression with limits and rationalization. This problem covers core calculus concepts suitable for Grades 11-12 and early college students.

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