The derivative of `f(x) = x^(4) e^(x)` is

Options

A. `4 x^(3) e^(x)`
B. `x^(4) e^(x)`
C. `x^(3) e^(x) (x + 4)`
D. `x^(2) e^(x) (x + 4)`

Answer

To find the derivative of the function \( f(x) = x^4 e^x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v' \]

### Step-by-step Solution:

1. **Identify the functions**:
Let \( u = x^4 \) and \( v = e^x \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) is:
\[ u' = \frac{d}{dx}(x^4) = 4x^3 \]
- The derivative of \( v \) is:
\[ v' = \frac{d}{dx}(e^x) = e^x \]

3. **Apply the product rule**:
Using the product rule, we have:
\[ f'(x) = u' \cdot v + u \cdot v' \]
Substituting the derivatives we found:
\[ f'(x) = (4x^3) \cdot (e^x) + (x^4) \cdot (e^x) \]

4. **Factor out common terms**:
Notice that both terms have \( e^x \) in common:
\[ f'(x) = e^x (4x^3 + x^4) \]

5. **Simplify the expression**:
We can rearrange the expression inside the parentheses:
\[ f'(x) = e^x x^3 (4 + x) \]

### Final Answer:
The derivative of \( f(x) = x^4 e^x \) is:
\[ f'(x) = e^x x^3 (4 + x) \]