Differentiate the following function w.r.t. x.
`f(x) = sqrt( 3 x + 4), x gt -1`

Answer

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

### Step 1: Rewrite the Function
We can express the square root in terms of an exponent:
\[ f(x) = (3x + 4)^{1/2} \]

**Hint:** Remember that \( \sqrt{a} = a^{1/2} \).

### Step 2: Apply the Chain Rule
To differentiate \( f(x) \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by:
\[ f'(x) = f'(g(x)) \cdot g'(x) \]
Here, let \( g(x) = 3x + 4 \) and \( f(g) = g^{1/2} \).

**Hint:** Identify the outer function and the inner function when using the chain rule.

### Step 3: Differentiate the Outer Function
Differentiate the outer function:
\[ \frac{d}{dx}(g^{1/2}) = \frac{1}{2} g^{-1/2} \]
Substituting back \( g(x) \):
\[ \frac{d}{dx}((3x + 4)^{1/2}) = \frac{1}{2}(3x + 4)^{-1/2} \]

**Hint:** Use the power rule for differentiation, which states that \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \).

### Step 4: Differentiate the Inner Function
Now, differentiate the inner function \( g(x) = 3x + 4 \):
\[ g'(x) = 3 \]

**Hint:** Remember that the derivative of a constant is zero.

### Step 5: Combine Using the Chain Rule
Now, apply the chain rule:
\[ f'(x) = \frac{1}{2}(3x + 4)^{-1/2} \cdot 3 \]

**Hint:** Multiply the derivatives of the outer and inner functions together.

### Step 6: Simplify the Expression
Now, simplify the expression:
\[ f'(x) = \frac{3}{2}(3x + 4)^{-1/2} \]
This can also be written as:
\[ f'(x) = \frac{3}{2\sqrt{3x + 4}} \]

**Hint:** Remember to express your final answer in a simplified form.

### Final Answer
Thus, the derivative of the function \( f(x) = \sqrt{3x + 4} \) with respect to \( x \) is:
\[ f'(x) = \frac{3}{2\sqrt{3x + 4}} \]