`e^(-2x)sin 4 x`

Answer

To find the differential coefficient of the function \( y = e^{-2x} \sin(4x) \) with respect to \( x \), we will use the product rule of differentiation.

### Step-by-step Solution:

1. **Identify the Functions**:
Let \( u = e^{-2x} \) and \( v = \sin(4x) \).

2. **Differentiate \( u \)**:
To find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \frac{d}{dx}(e^{-2x}) = e^{-2x} \cdot (-2) = -2e^{-2x} \]

3. **Differentiate \( v \)**:
To find \( \frac{dv}{dx} \):
\[ \frac{dv}{dx} = \frac{d}{dx}(\sin(4x)) = \cos(4x) \cdot 4 = 4\cos(4x) \]

4. **Apply the Product Rule**:
The product rule states that:
\[ \frac{d}{dx}(u \cdot v) = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} \]
Substituting the values we have:
\[ \frac{dy}{dx} = e^{-2x} \cdot (4\cos(4x)) + \sin(4x) \cdot (-2e^{-2x}) \]

5. **Simplify the Expression**:
\[ \frac{dy}{dx} = 4e^{-2x}\cos(4x) - 2e^{-2x}\sin(4x) \]

6. **Factor Out Common Terms**:
We can factor out \( 2e^{-2x} \):
\[ \frac{dy}{dx} = 2e^{-2x}(2\cos(4x) - \sin(4x)) \]

### Final Answer:
Thus, the differential coefficient of the function \( y = e^{-2x} \sin(4x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2e^{-2x}(2\cos(4x) - \sin(4x)) \]