If `y=e^(1+log_(e)x)`, then the value of `(dy)/(dx)` is equal to

Options

A. e
B. 1
C. 0
D. `log_(e)xelog_(e^(e^(x)))`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = e^{1 + \log_e x} \), we can follow these steps:

### Step 1: Simplify the expression for \( y \)
We start with the given function:
\[ y = e^{1 + \log_e x} \]
Using the property of exponents, we can rewrite this as:
\[ y = e^1 \cdot e^{\log_e x} = e \cdot x \]

### Step 2: Differentiate \( y \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(e \cdot x) \]
Since \( e \) is a constant, the derivative is:
\[ \frac{dy}{dx} = e \cdot \frac{d}{dx}(x) = e \cdot 1 = e \]

### Final Result
Thus, the value of \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = e \]