Question

If \( f(x) = e^{x}g(x) \), \( g(0) = 4 \), and \( g'(0) = 2 \), then \( f'(0) = \)

• A. 4
• B. 6
• C. 1
• D. 2

Hint:

When differentiating products, remember to apply the product rule: \( (uv)' = u'v + uv' \).

Answer 1

The Correct Option is B

Step 1: Apply the product rule.
We are given that \( f(x) = e^x g(x) \). To find \( f'(x) \), we use the product rule: \[ f'(x) = \frac{d}{dx} \left( e^x \right) g(x) + e^x \frac{d}{dx} \left( g(x) \right) \] This simplifies to: \[ f'(x) = e^x g(x) + e^x g'(x) \] Step 2: Evaluate at \( x = 0 \).
Now, substituting the known values at \( x = 0 \): \[ f'(0) = e^0 \cdot g(0) + e^0 \cdot g'(0) = 1 \cdot 4 + 1 \cdot 2 = 6 \] Step 3: Conclusion.
Thus, the value of \( f'(0) \) is \( \boxed{6} \).