If f, of, x, equals, 3, x, e, start superscript, minus, x, end superscript, commaf(x)=3xe 
−x
 , then what is the value of f, prime, of, 2, question markf 
′
 (2)? Express your answer in simplest form with no negative exponents.

To find \( f'(2) \), we first need to determine the derivative of the function \( f(x) = 3x e^{-x} \).

### Step 1: Differentiate \( f(x) = 3x e^{-x} \)

To find \( f'(x) \), we’ll use the **product rule** for differentiation. Recall that if \( u(x) \) and \( v(x) \) are functions of \( x \), then the derivative of their product is given by:
\[
(uv)' = u'v + uv'
\]

In this case, let:
- \( u = 3x \) and \( u' = 3 \)
- \( v = e^{-x} \) and \( v' = -e^{-x} \) (since the derivative of \( e^{-x} \) is \( -e^{-x} \))

Now, applying the product rule:
\[
f'(x) = u'v + uv' = (3)(e^{-x}) + (3x)(-e^{-x})
\]
Simplifying this, we get:
\[
f'(x) = 3e^{-x} - 3x e^{-x}
\]
or equivalently,
\[
f'(x) = e^{-x}(3 - 3x)
\]

### Step 2: Substitute \( x = 2 \) into \( f'(x) \)

Now we need to find \( f'(2) \):
\[
f'(2) = e^{-2}(3 - 3 \cdot 2)
\]
\[
= e^{-2}(3 - 6)
\]
\[
= e^{-2}(-3)
\]
\[
= -3e^{-2}
\]

### Step 3: Simplify the answer with no negative exponents

Since we are asked to express the answer without negative exponents:
\[
f'(2) = -\frac{3}{e^2}
\]

### Final Answer:
\[
f'(2) = -\frac{3}{e^2}
\]

Would you like more details on any part of this solution?

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Here are 5 related questions that might interest you:
1. How is the product rule applied in differentiation?
2. How do you find the derivative of functions involving \( e^x \) terms?
3. What would be \( f'(x) \) if \( f(x) = 5x e^{2x} \)?
4. How would you find the second derivative \( f''(x) \) of \( f(x) = 3x e^{-x} \)?
5. Why is \( e^x \) its own derivative, and what is unique about exponential functions?

**Tip:** When differentiating products, always identify each component and use the product rule step-by-step to avoid mistakes.

If f(x) = 3x e^{-x}, then what is the value of f'(2)? Express your answer in simplest form with no negative exponents.

Learn how to find the derivative of f(x) = 3x e^{-x} and evaluate f'(2) using the product rule. This solution involves calculus concepts, differentiation techniques, and simplifying exponential terms. Ideal for high school calculus students.

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