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

Options

A. ` (2-x)/( x) `
B. ` (x-2)/( x) `
C. ` (e-x)/( xe) `
D. ` (x-e)/(ex) `

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = \log\left(\frac{e^x}{x^2}\right) \), we can follow these steps:

### Step 1: Simplify the logarithmic expression
Using the properties of logarithms, we can rewrite the expression:
\[ y = \log\left(\frac{e^x}{x^2}\right) = \log(e^x) - \log(x^2) \]
Using the property \( \log(a/b) = \log(a) - \log(b) \) and \( \log(a^b) = b \log(a) \), we can further simplify:
\[ y = x - 2\log(x) \]

### Step 2: Differentiate the expression
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x) - \frac{d}{dx}(2\log(x)) \]
The derivative of \( x \) is \( 1 \), and the derivative of \( \log(x) \) is \( \frac{1}{x} \):
\[ \frac{dy}{dx} = 1 - 2 \cdot \frac{1}{x} \]

### Step 3: Simplify the derivative
Now we simplify the expression:
\[ \frac{dy}{dx} = 1 - \frac{2}{x} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 1 - \frac{2}{x} \]

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