If ` y=( log x )^(x) -x^(log x) ,then (dy)/(dx)=`

Options

A. ` (log x )^(x) ((1)/( logx ) + log (log x ) )+ ((log x)x^(log x))/( x ) `
B. ` (log x )^(x) ((1)/( logx ) + log (log x ) )-((log x)x^(log x))/( x ) `
C. ` (log x )^(x) ((1)/( logx ) + log (log x ) )+ (2(log x)x^(log x))/( x ) `
D. ` (log x )^(x) ((1)/( logx ) + log (log x ) )- (2(log x)x^(log x))/( x ) `

Answer

To find the derivative of the function \( y = (\log x)^x - x^{\log x} \), we will differentiate each term separately.

### Step 1: Differentiate \( y = (\log x)^x \)

Let \( a = (\log x)^x \).

Taking the logarithm of both sides:
\[ \log a = x \log(\log x) \]

Now, differentiate both sides with respect to \( x \):
\[ \frac{1}{a} \frac{da}{dx} = \log(\log x) + x \cdot \frac{1}{\log x} \cdot \frac{1}{x} \]
Here, we used the product rule for differentiation on \( x \log(\log x) \).

This simplifies to:
\[ \frac{1}{a} \frac{da}{dx} = \log(\log x) + \frac{1}{\log x} \]

Now, multiplying both sides by \( a \):
\[ \frac{da}{dx} = a \left( \log(\log x) + \frac{1}{\log x} \right) \]

Substituting back \( a = (\log x)^x \):
\[ \frac{da}{dx} = (\log x)^x \left( \log(\log x) + \frac{1}{\log x} \right) \]

### Step 2: Differentiate \( y = x^{\log x} \)

Let \( b = x^{\log x} \).

Taking the logarithm of both sides:
\[ \log b = \log x \cdot \log x = (\log x)^2 \]

Now, differentiate both sides with respect to \( x \):
\[ \frac{1}{b} \frac{db}{dx} = 2 \log x \cdot \frac{1}{x} \]

This simplifies to:
\[ \frac{1}{b} \frac{db}{dx} = \frac{2 \log x}{x} \]

Now, multiplying both sides by \( b \):
\[ \frac{db}{dx} = b \cdot \frac{2 \log x}{x} \]

Substituting back \( b = x^{\log x} \):
\[ \frac{db}{dx} = x^{\log x} \cdot \frac{2 \log x}{x} = 2 \log x \cdot x^{\log x - 1} \]

### Step 3: Combine the derivatives

Now we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{da}{dx} - \frac{db}{dx} \]
Substituting the expressions we found:
\[ \frac{dy}{dx} = (\log x)^x \left( \log(\log x) + \frac{1}{\log x} \right) - 2 \log x \cdot x^{\log x - 1} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = (\log x)^x \left( \log(\log x) + \frac{1}{\log x} \right) - 2 \log x \cdot x^{\log x - 1} \]