Differentiate the following w.r.t.x. `e^(x)sinx`

Answer

To differentiate the function \( y = e^x \sin x \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions multiplied together, say \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}(u \cdot v) = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step-by-step Solution:

1. **Identify the functions**:
Let \( u = e^x \) and \( v = \sin x \).

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = e^x \]
- The derivative of \( v \) with respect to \( x \) is:
\[ \frac{dv}{dx} = \cos x \]

3. **Apply the product rule**:
Using the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = e^x \cos x + \sin x \cdot e^x \]

4. **Factor out common terms**:
Notice that \( e^x \) is common in both terms:
\[ \frac{dy}{dx} = e^x (\cos x + \sin x) \]

5. **Final result**:
Thus, the derivative of \( y = e^x \sin x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = e^x (\cos x + \sin x) \]