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The question asks to determine the instantaneous rate of change of the function \( f(x) = \sqrt{x} - 2 \) at \( x = 6 \), using the limit definition of the derivative:

\[
\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Let's break down the solution:

### Step 1: Plug the function \( f(x) = \sqrt{x} - 2 \) into the limit definition of the derivative.
For \( x = 6 \), the expression becomes:

\[
\lim_{h \to 0} \frac{f(6+h) - f(6)}{h}
\]

### Step 2: Compute \( f(6) \)
Using \( f(x) = \sqrt{x} - 2 \):

\[
f(6) = \sqrt{6} - 2
\]

### Step 3: Compute \( f(6+h) \)
Now substitute \( x = 6+h \) into the function:

\[
f(6+h) = \sqrt{6+h} - 2
\]

### Step 4: Substitute into the limit expression
Now substitute \( f(6+h) \) and \( f(6) \) into the limit definition:

\[
\lim_{h \to 0} \frac{\left( \sqrt{6+h} - 2 \right) - \left( \sqrt{6} - 2 \right)}{h}
\]

Simplifying the numerator:

\[
\frac{\sqrt{6+h} - 2 - \sqrt{6} + 2}{h} = \frac{\sqrt{6+h} - \sqrt{6}}{h}
\]

### Step 5: Rationalize the numerator
To simplify this limit, rationalize the numerator by multiplying the numerator and denominator by the conjugate:

\[
\lim_{h \to 0} \frac{\sqrt{6+h} - \sqrt{6}}{h} \cdot \frac{\sqrt{6+h} + \sqrt{6}}{\sqrt{6+h} + \sqrt{6}}
\]

This simplifies to:

\[
\lim_{h \to 0} \frac{(6+h) - 6}{h(\sqrt{6+h} + \sqrt{6})} = \lim_{h \to 0} \frac{h}{h(\sqrt{6+h} + \sqrt{6})}
\]

Cancel \( h \) from the numerator and denominator:

\[
\lim_{h \to 0} \frac{1}{\sqrt{6+h} + \sqrt{6}}
\]

### Step 6: Evaluate the limit
As \( h \to 0 \), \( \sqrt{6+h} \to \sqrt{6} \). Therefore, the limit becomes:

\[
\frac{1}{2\sqrt{6}}
\]

This is the instantaneous rate of change of the function at \( x = 6 \).

Would you like to explore more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you compute derivatives using the chain rule?
2. What are some alternative methods to find the derivative of square root functions?
3. Can you explain the geometric interpretation of the derivative as a slope?
4. How would the calculation differ for other types of functions, like \( f(x) = x^2 - 2x \)?
5. How can we extend this method to calculate second derivatives?

**Tip:** Always remember to rationalize the numerator when dealing with square root functions in limits!

Given the function f(x) = √x - 2, determine the instantaneous rate of change of f at x = 6 using the limit shown below.

Determine the instantaneous rate of change of f(x) = √x - 2 at x = 6 using the limit definition of the derivative. This problem involves limits, rationalization techniques, and understanding how to compute derivatives of square root functions. Suitable for students in Grades 10-12.

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