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We are asked to find the derivative of the function \( f(x) = \sqrt{7 - 4x} \) using the limit definition of the derivative.

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

Let's apply this to \( f(x) = \sqrt{7 - 4x} \).

1. First, compute \( f(x + h) \):
   \[
   f(x + h) = \sqrt{7 - 4(x + h)} = \sqrt{7 - 4x - 4h}
   \]

2. Now, substitute this into the limit definition:
   \[
   f'(x) = \lim_{h \to 0} \frac{\sqrt{7 - 4x - 4h} - \sqrt{7 - 4x}}{h}
   \]

3. To simplify, we rationalize the numerator by multiplying the numerator and denominator by the conjugate:
   \[
   f'(x) = \lim_{h \to 0} \frac{\left( \sqrt{7 - 4x - 4h} - \sqrt{7 - 4x} \right) \cdot \left( \sqrt{7 - 4x - 4h} + \sqrt{7 - 4x} \right)}{h \cdot \left( \sqrt{7 - 4x - 4h} + \sqrt{7 - 4x} \right)}
   \]

4. The numerator simplifies using the difference of squares:
   \[
   = \lim_{h \to 0} \frac{(7 - 4x - 4h) - (7 - 4x)}{h \left( \sqrt{7 - 4x - 4h} + \sqrt{7 - 4x} \right)}
   \]
   Which simplifies further to:
   \[
   = \lim_{h \to 0} \frac{-4h}{h \left( \sqrt{7 - 4x - 4h} + \sqrt{7 - 4x} \right)}
   \]

5. Cancel out the \( h \) in the numerator and denominator:
   \[
   = \lim_{h \to 0} \frac{-4}{\sqrt{7 - 4x - 4h} + \sqrt{7 - 4x}}
   \]

6. Finally, take the limit as \( h \to 0 \):
   \[
   f'(x) = \frac{-4}{2\sqrt{7 - 4x}} = \frac{-2}{\sqrt{7 - 4x}}
   \]

Thus, the derivative of the function is:
\[
f'(x) = \frac{-2}{\sqrt{7 - 4x}}
\]

Would you like more details on any step?

Here are some related questions:
1. How can we interpret the result geometrically?
2. Can we use the chain rule to verify this derivative?
3. What happens to \( f'(x) \) as \( x \) approaches a value where the function becomes undefined?
4. How would we apply this technique to a more complex radical function?
5. How does rationalizing the numerator help simplify the limit?

**Tip:** Rationalizing expressions with square roots is a key step in simplifying many limit problems.

Use the limit definition of derivative to find f'(x), where f(x) = √(7 - 4x).

Learn how to use the limit definition of the derivative to find f'(x) for the function f(x) = √(7 - 4x). This detailed solution covers key calculus concepts, including limits, derivatives, and rationalizing square roots. Suitable for high school or Calculus 1 students.

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