If ` y = e^(x+2log x ),then (dy)/(dx)=`

Options

A. ` x(x^(2) +2) e^(x) `
B. ` x(x^(2) +2) e^(2x) `
C. ` x(x+2) e^(x) `
D. ` x( x+2) e^(2x) `

Answer

To find the derivative of the function \( y = e^{x + 2 \log x} \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

### Step 1: Identify the function and its components
We have:
\[ y = e^{x + 2 \log x} \]
Let \( f(x) = x + 2 \log x \). Thus, we can rewrite \( y \) as:
\[ y = e^{f(x)} \]

### Step 2: Differentiate using the chain rule
The derivative of \( y \) with respect to \( x \) using the chain rule is given by:
\[ \frac{dy}{dx} = e^{f(x)} \cdot \frac{df}{dx} \]

### Step 3: Differentiate \( f(x) \)
Now, we need to find \( \frac{df}{dx} \):
\[ f(x) = x + 2 \log x \]
Differentiating \( f(x) \):
\[ \frac{df}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}(2 \log x) \]
The derivative of \( x \) is \( 1 \), and the derivative of \( 2 \log x \) is:
\[ \frac{d}{dx}(2 \log x) = 2 \cdot \frac{1}{x} = \frac{2}{x} \]
Thus,
\[ \frac{df}{dx} = 1 + \frac{2}{x} \]

### Step 4: Substitute back into the derivative of \( y \)
Now substituting \( f(x) \) and \( \frac{df}{dx} \) back into the derivative of \( y \):
\[ \frac{dy}{dx} = e^{f(x)} \cdot \left(1 + \frac{2}{x}\right) \]
Substituting \( f(x) = x + 2 \log x \):
\[ \frac{dy}{dx} = e^{x + 2 \log x} \cdot \left(1 + \frac{2}{x}\right) \]

### Step 5: Simplify the expression
We can simplify \( e^{x + 2 \log x} \):
Using the property of logarithms, \( 2 \log x = \log x^2 \), we have:
\[ e^{x + 2 \log x} = e^{x} \cdot e^{\log x^2} = e^{x} \cdot x^2 \]
Thus:
\[ \frac{dy}{dx} = e^{x} \cdot x^2 \cdot \left(1 + \frac{2}{x}\right) \]
Now simplifying \( 1 + \frac{2}{x} \):
\[ 1 + \frac{2}{x} = \frac{x + 2}{x} \]
So we have:
\[ \frac{dy}{dx} = e^{x} \cdot x^2 \cdot \frac{x + 2}{x} = e^{x} \cdot \frac{x^2(x + 2)}{x} = e^{x} \cdot (x + 2) \cdot x \]

### Final Answer
Thus, the final answer is:
\[ \frac{dy}{dx} = e^{x} \cdot (x + 2) \]