If `y=sin(x)+"in"(x^(2))+e^(2x)"then"(dy)/(dx)` will be :

Options

A. `cosx+(2)/(x)+e^(2x)`
B. `cos x+(2)/(x)+2e^(2x)`
C. `-cos x+(2)/(x^(2))+e^(2x)`
D. `-ocs x-(2)/(x^(2))+2e^(2x)`

Explanation

To find the derivative of the function \( y = \sin(x) + \ln(x^2) + e^{2x} \), we will differentiate each term separately and then combine the results.

### Step-by-Step Solution:

1. **Differentiate \( \sin(x) \)**:
\[ \frac{d}{dx}(\sin(x)) = \cos(x) \]

2. **Differentiate \( \ln(x^2) \)**:
- First, we apply the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).
- Here, \( u = x^2 \), so:
\[ \frac{d}{dx}(\ln(x^2)) = \frac{1}{x^2} \cdot \frac{d}{dx}(x^2) = \frac{1}{x^2} \cdot 2x = \frac{2}{x} \]

3. **Differentiate \( e^{2x} \)**:
- Using the chain rule again, the derivative of \( e^{u} \) is \( e^{u} \cdot \frac{du}{dx} \).
- Here, \( u = 2x \), so:
\[ \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2 = 2e^{2x} \]

4. **Combine the derivatives**:
- Now, we can combine all the derivatives we calculated:
\[ \frac{dy}{dx} = \cos(x) + \frac{2}{x} + 2e^{2x} \]

5. **Final Result**:
- Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \cos(x) + \frac{2}{x} + 2e^{2x} \]