Differentiate the following functions:
`x^(3) + 4x^(2) + 7x +2`

Answer

To differentiate the function \( f(x) = x^3 + 4x^2 + 7x + 2 \), we will apply the rules of differentiation step by step.

### Step 1: Identify the function
The function we need to differentiate is:
\[ f(x) = x^3 + 4x^2 + 7x + 2 \]

### Step 2: Apply the differentiation rules
We will use the following differentiation rules:
1. Power Rule: \( \frac{d}{dx}(x^m) = m \cdot x^{m-1} \)
2. Constant Multiple Rule: \( \frac{d}{dx}(k \cdot f(x)) = k \cdot \frac{d}{dx}(f(x)) \)
3. Constant Rule: \( \frac{d}{dx}(k) = 0 \)

### Step 3: Differentiate each term
Now we will differentiate each term of the function separately.

1. Differentiate \( x^3 \):
\[ \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 \]

2. Differentiate \( 4x^2 \):
\[ \frac{d}{dx}(4x^2) = 4 \cdot \frac{d}{dx}(x^2) = 4 \cdot 2x^{2-1} = 8x \]

3. Differentiate \( 7x \):
\[ \frac{d}{dx}(7x) = 7 \cdot \frac{d}{dx}(x) = 7 \cdot 1 = 7 \]

4. Differentiate the constant \( 2 \):
\[ \frac{d}{dx}(2) = 0 \]

### Step 4: Combine the results
Now we will combine all the differentiated terms:
\[ f'(x) = 3x^2 + 8x + 7 + 0 \]
Thus, we can simplify this to:
\[ f'(x) = 3x^2 + 8x + 7 \]

### Final Result
The derivative of the function \( f(x) = x^3 + 4x^2 + 7x + 2 \) is:
\[ f'(x) = 3x^2 + 8x + 7 \]

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