Differentiate the following w.r.t.x. `(x^(2)+x+1)^(4)`

Answer

To differentiate the function \( (x^2 + x + 1)^4 \) with respect to \( x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
Let:
\[ y = (x^2 + x + 1)^4 \]

### Step 2: Differentiate using the chain rule
According to the chain rule, if \( y = u^n \), where \( u = x^2 + x + 1 \) and \( n = 4 \), then:
\[ \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \]

### Step 3: Differentiate the outer function
Here, \( n = 4 \) and \( u = x^2 + x + 1 \):
\[ \frac{dy}{dx} = 4 \cdot (x^2 + x + 1)^{3} \cdot \frac{du}{dx} \]

### Step 4: Differentiate the inner function
Now, we need to find \( \frac{du}{dx} \):
\[ u = x^2 + x + 1 \]
Differentiating \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2) + \frac{d}{dx}(x) + \frac{d}{dx}(1) \]
\[ \frac{du}{dx} = 2x + 1 + 0 \]
\[ \frac{du}{dx} = 2x + 1 \]

### Step 5: Substitute back into the derivative
Now, substituting \( \frac{du}{dx} \) back into the equation for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 4 \cdot (x^2 + x + 1)^{3} \cdot (2x + 1) \]

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

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