If `y="log"(xe^(x)),"then "(dy)/(dx)` is

Options

A. `(x+1)/(x)`
B. `1/(x)`
C. `1-x`
D. none of the above

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = \log(x e^x) \), we will follow these steps:

### Step 1: Rewrite the function using logarithmic properties
We can use the property of logarithms that states \( \log(ab) = \log(a) + \log(b) \). Therefore, we can rewrite the function as:
\[ y = \log(x) + \log(e^x) \]
Since \( \log(e^x) = x \), we can simplify it further:
\[ y = \log(x) + x \]

### Step 2: Differentiate \( y \) with respect to \( x \)
Now, we will differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\log(x)) + \frac{d}{dx}(x) \]
Using the derivatives of logarithmic and polynomial functions:
\[ \frac{dy}{dx} = \frac{1}{x} + 1 \]

### Step 3: Combine the terms
Now we can combine the terms:
\[ \frac{dy}{dx} = \frac{1}{x} + 1 = \frac{1 + x}{x} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{x + 1}{x} \]

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