Question

The derivative of $e^{2x}\sin x$ with respect to $x$ is

• A. $e^{2x}(\cos x + 2\sin x)$
• B. $e^{2x}(\sin x + \cos x)$
• C. $2e^{2x}\cos x$
• D. $e^{x}\sin(2x)$

Hint:

Always use the product rule when differentiating the product of two functions.

Answer 1

The Correct Option is A

Step 1: Apply the product rule.
If $y = uv$, then \[ \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} \]
Step 2: Identify functions.
Let $u = e^{2x}$ and $v = \sin x$.
Step 3: Differentiate each part.
\[ \frac{du}{dx} = 2e^{2x}, \quad \frac{dv}{dx} = \cos x \]
Step 4: Substitute in product rule.
\[ \frac{d}{dx}(e^{2x}\sin x) = e^{2x}\cos x + \sin x \cdot 2e^{2x} \] \[ = e^{2x}(\cos x + 2\sin x) \]