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 \( \frac{dy}{dx} \) for the function given by

\[ y = \sin(x) + \ln(x^2) + e^{2x}, \]

we will differentiate each term separately.

### Step 1: Differentiate \( \sin(x) \)

The derivative of \( \sin(x) \) is

\[ \frac{d}{dx}(\sin(x)) = \cos(x). \]

### Step 2: Differentiate \( \ln(x^2) \)

Using the property of logarithms that states \( \ln(a^b) = b \ln(a) \), we can rewrite \( \ln(x^2) \) as

\[ \ln(x^2) = 2\ln(x). \]

Now, we differentiate \( 2\ln(x) \):

\[ \frac{d}{dx}(2\ln(x)) = 2 \cdot \frac{1}{x} = \frac{2}{x}. \]

### Step 3: Differentiate \( e^{2x} \)

Using the chain rule, the derivative of \( e^{2x} \) is

\[ \frac{d}{dx}(e^{2x}) = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2 = 2e^{2x}. \]

### Step 4: Combine the derivatives

Now, we combine all the derivatives we found:

\[ \frac{dy}{dx} = \cos(x) + \frac{2}{x} + 2e^{2x}. \]

### Final Answer

Thus, the derivative \( \frac{dy}{dx} \) is

\[ \frac{dy}{dx} = \cos(x) + \frac{2}{x} + 2e^{2x}. \]

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