Use the limit process to find the derivative of the function.
f(x) = 

√x + 2
f ′(x)	 = 	
lim 
Δx→0 
 
f(x + Δx − f(x))
Δx
 = 	
lim 
Δx→0 
  − 


√x + 2

Δx
 = 	
 

To find the derivative of the function \( f(x) = \sqrt{x + 2} \) using the limit definition of the derivative, let's walk through the steps clearly.

### Step 1: Recall the definition of the derivative
The derivative of \( f(x) \) at any point \( x \) is given by:

\[
f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
\]

### Step 2: Plug in the function \( f(x) = \sqrt{x + 2} \)

We substitute \( f(x) \) into the definition:

\[
f'(x) = \lim_{\Delta x \to 0} \frac{\sqrt{(x + \Delta x) + 2} - \sqrt{x + 2}}{\Delta x}
\]

This simplifies to:

\[
f'(x) = \lim_{\Delta x \to 0} \frac{\sqrt{x + \Delta x + 2} - \sqrt{x + 2}}{\Delta x}
\]

### Step 3: Multiply by the conjugate

To handle the square roots in the numerator, we multiply both the numerator and denominator by the conjugate of the numerator:

\[
f'(x) = \lim_{\Delta x \to 0} \frac{\left(\sqrt{x + \Delta x + 2} - \sqrt{x + 2}\right) \left(\sqrt{x + \Delta x + 2} + \sqrt{x + 2}\right)}{\Delta x \left(\sqrt{x + \Delta x + 2} + \sqrt{x + 2}\right)}
\]

Multiplying the numerator:

\[
\left(\sqrt{x + \Delta x + 2}\right)^2 - \left(\sqrt{x + 2}\right)^2 = \left(x + \Delta x + 2\right) - \left(x + 2\right) = \Delta x
\]

So, the expression becomes:

\[
f'(x) = \lim_{\Delta x \to 0} \frac{\Delta x}{\Delta x \left(\sqrt{x + \Delta x + 2} + \sqrt{x + 2}\right)}
\]

### Step 4: Simplify and take the limit

Now, cancel out \( \Delta x \) from the numerator and denominator:

\[
f'(x) = \lim_{\Delta x \to 0} \frac{1}{\sqrt{x + \Delta x + 2} + \sqrt{x + 2}}
\]

As \( \Delta x \to 0 \), \( \sqrt{x + \Delta x + 2} \to \sqrt{x + 2} \), so we can substitute this into the limit:

\[
f'(x) = \frac{1}{2\sqrt{x + 2}}
\]

### Final Answer:

The derivative of \( f(x) = \sqrt{x + 2} \) is:

\[
f'(x) = \frac{1}{2\sqrt{x + 2}}
\]

---

Would you like further explanation on any of these steps? 

Here are 5 related questions to explore this further:
1. What is the derivative of \( f(x) = \sqrt{x + 1} \) using the same process?
2. How would you find the derivative of \( f(x) = \frac{1}{x + 2} \) using the limit process?
3. How do conjugates help simplify expressions with square roots?
4. Can we use this limit process for any differentiable function?
5. What is the geometrical interpretation of finding a derivative using limits?

**Tip**: When dealing with square roots in the numerator, multiplying by the conjugate is a common technique to simplify the expression.

Use the limit process to find the derivative of the function f(x) = √x + 2.

Learn how to find the derivative of the function f(x) = √x + 2 using the limit definition. This solution walks through key steps including applying the conjugate multiplication technique to simplify square roots, leading to the final derivative f'(x) = 1 / 2√(x + 2). Ideal for students in Grades 11-12 studying calculus.

Math Problem Statement

f(x + Δx − f(x)) Δx

Δx

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