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

Options

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

Answer

To find the derivative of the function \( f(x) = \frac{x^4}{e^x} \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

where \( u = x^4 \) and \( v = e^x \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x^4 \)
- \( v = e^x \)

### Step 2: Find \( u' \) and \( v' \)
Now, we need to find the derivatives of \( u \) and \( v \):
- \( u' = \frac{d}{dx}(x^4) = 4x^3 \)
- \( v' = \frac{d}{dx}(e^x) = e^x \)

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

Substituting \( u \), \( u' \), \( v \), and \( v' \):

\[ f'(x) = \frac{(4x^3)(e^x) - (x^4)(e^x)}{(e^x)^2} \]

### Step 4: Simplify the Expression
We can factor out \( e^x \) from the numerator:

\[ f'(x) = \frac{e^x(4x^3 - x^4)}{(e^x)^2} \]

This simplifies to:

\[ f'(x) = \frac{4x^3 - x^4}{e^x} \]

### Step 5: Factor the Numerator
We can factor \( x^3 \) out of the numerator:

\[ f'(x) = \frac{x^3(4 - x)}{e^x} \]

### Final Answer
Thus, the derivative of \( f(x) = \frac{x^4}{e^x} \) is:

\[ f'(x) = \frac{x^3(4 - x)}{e^x} \]

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