Find the derivative of `|x^(2)+2|` w.r.t.x.

Answer

To find the derivative of the function \( y = |x^2 + 2| \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function inside the modulus
The function inside the modulus is \( u = x^2 + 2 \).

### Step 2: Determine the derivative of the modulus function
The derivative of the modulus function \( |u| \) can be expressed using the formula:
\[ \frac{dy}{dx} = \frac{u}{|u|} \cdot \frac{du}{dx} \]
where \( u = x^2 + 2 \).

### Step 3: Calculate \( \frac{du}{dx} \)
Now, we need to find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 2) = 2x \]

### Step 4: Substitute back into the derivative formula
Now we can substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula:
\[ \frac{dy}{dx} = \frac{x^2 + 2}{|x^2 + 2|} \cdot 2x \]

### Step 5: Simplify the expression
Since \( x^2 + 2 \) is always positive for all real \( x \) (as the minimum value of \( x^2 \) is 0, making \( x^2 + 2 \geq 2 \)), we have:
\[ |x^2 + 2| = x^2 + 2 \]
Thus, the derivative simplifies to:
\[ \frac{dy}{dx} = \frac{x^2 + 2}{x^2 + 2} \cdot 2x = 2x \]

### Final Answer
The derivative of \( |x^2 + 2| \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2x \]

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