Differentiate the following w.r.t. x :
`(x^(2)+7x+2)(e^(x)-sinx)`

Answer

To differentiate the function \( y = (x^2 + 7x + 2)(e^x - \sin x) \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:

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

### Step-by-step Solution:

1. **Identify the functions**:
Let
\[ u = x^2 + 7x + 2 \]
and
\[ v = e^x - \sin x \]

2. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 7x + 2) = 2x + 7 \]

3. **Differentiate \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(e^x - \sin x) = e^x - \cos x \]

4. **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 \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = (x^2 + 7x + 2)(e^x - \cos x) + (e^x - \sin x)(2x + 7) \]

5. **Simplify the expression**:
The final expression for the derivative is:
\[ \frac{dy}{dx} = (x^2 + 7x + 2)(e^x - \cos x) + (e^x - \sin x)(2x + 7) \]

### Final Answer:
\[ \frac{dy}{dx} = (x^2 + 7x + 2)(e^x - \cos x) + (e^x - \sin x)(2x + 7) \]