Differentiate `(x^(2)+3x+4)^(n),n inI` w.r.t. x.

Answer

To differentiate the function \( (x^2 + 3x + 4)^n \) with respect to \( x \), we will use the chain rule. Here are the steps to solve the problem:

### Step 1: Identify the outer and inner functions
Let \( y = (x^2 + 3x + 4)^n \). Here, the outer function is \( u^n \) where \( u = x^2 + 3x + 4 \).

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = n \cdot (u)^{n-1} \cdot \frac{du}{dx} \]

### Step 3: Differentiate the inner function
Now we need to find \( \frac{du}{dx} \):
\[ u = x^2 + 3x + 4 \]
Differentiating \( u \) with respect to \( x \):
\[ \frac{du}{dx} = 2x + 3 \]

### Step 4: Substitute back into the derivative
Now substitute \( u \) and \( \frac{du}{dx} \) back into the derivative:
\[ \frac{dy}{dx} = n \cdot (x^2 + 3x + 4)^{n-1} \cdot (2x + 3) \]

### Final Answer
Thus, the derivative of \( (x^2 + 3x + 4)^n \) with respect to \( x \) is:
\[ \frac{dy}{dx} = n \cdot (x^2 + 3x + 4)^{n-1} \cdot (2x + 3) \]