Differentiable the following w.r.t. x :
`sqrt(3x+2)+1/(sqrt(2x^(2)+4))`

Answer

To differentiate the function \( y = \sqrt{3x + 2} + \frac{1}{\sqrt{2x^2 + 4}} \) with respect to \( x \), we will follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function in a more convenient form for differentiation:
\[ y = (3x + 2)^{1/2} + (2x^2 + 4)^{-1/2} \]

### Step 2: Differentiate using the chain rule
We will differentiate each term separately using the chain rule.

1. **Differentiate the first term**:
\[ \frac{d}{dx}[(3x + 2)^{1/2}] = \frac{1}{2}(3x + 2)^{-1/2} \cdot \frac{d}{dx}(3x + 2) \]
The derivative of \( 3x + 2 \) is \( 3 \), so:
\[ \frac{d}{dx}[(3x + 2)^{1/2}] = \frac{1}{2}(3x + 2)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x + 2}} \]

2. **Differentiate the second term**:
\[ \frac{d}{dx}[(2x^2 + 4)^{-1/2}] = -\frac{1}{2}(2x^2 + 4)^{-3/2} \cdot \frac{d}{dx}(2x^2 + 4) \]
The derivative of \( 2x^2 + 4 \) is \( 4x \), so:
\[ \frac{d}{dx}[(2x^2 + 4)^{-1/2}] = -\frac{1}{2}(2x^2 + 4)^{-3/2} \cdot 4x = -\frac{2x}{(2x^2 + 4)^{3/2}} \]

### Step 3: Combine the derivatives
Now we can combine the derivatives of both terms:
\[ \frac{dy}{dx} = \frac{3}{2\sqrt{3x + 2}} - \frac{2x}{(2x^2 + 4)^{3/2}} \]

### Final Answer
Thus, the derivative of the given function with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{3}{2\sqrt{3x + 2}} - \frac{2x}{(2x^2 + 4)^{3/2}} \]