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 solve the problem, we need to differentiate the function \( y = e^{1 + \log_e x} \). Let's go through the steps to find \( \frac{dy}{dx} \).

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

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

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

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