if ` y= a^(x^(a^(x. . ."" oo)))` , then ` x(1-y log x log y) (dy)/(dx)=`

Options

A. `y^2 log y `
B. ` y log y `
C. ` y^2 //log y `
D. None

Answer

To solve the problem \( y = a^{x^{a^{x^{\cdots}}}} \), we will first express the infinite exponentiation in a manageable form, differentiate, and then manipulate the resulting equation to find the required expression.

### Step-by-Step Solution:

1. **Express the Infinite Exponentiation**:
We start with the equation:
\[ y = a^{x^{a^{x^{\cdots}}}} \]
Since the exponent itself is \( y \), we can rewrite this as:
\[ y = a^{x^y} \]

2. **Take the Natural Logarithm**:
To simplify the differentiation, we take the logarithm of both sides:
\[ \log y = x^y \log a \]

3. **Differentiate Both Sides**:
Now we differentiate both sides with respect to \( x \). Using implicit differentiation:
- The left-hand side (LHS) requires the chain rule:
\[ \frac{d}{dx}(\log y) = \frac{1}{y} \frac{dy}{dx} \]
- The right-hand side (RHS) requires the product rule:
\[ \frac{d}{dx}(x^y \log a) = \log a \left( y x^{y-1} + x^y \frac{dy}{dx} \log x \right) \]

4. **Set the Derivatives Equal**:
Now we can set the derivatives equal to each other:
\[ \frac{1}{y} \frac{dy}{dx} = \log a \left( y x^{y-1} + x^y \frac{dy}{dx} \log x \right) \]

5. **Isolate \(\frac{dy}{dx}\)**:
Rearranging gives:
\[ \frac{1}{y} \frac{dy}{dx} - \log a \cdot x^y \log x \cdot \frac{dy}{dx} = \log a \cdot y x^{y-1} \]
Factor out \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} \left( \frac{1}{y} - \log a \cdot x^y \log x \right) = \log a \cdot y x^{y-1} \]

6. **Solve for \(\frac{dy}{dx}\)**:
Thus, we can express \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = \frac{\log a \cdot y x^{y-1}}{\frac{1}{y} - \log a \cdot x^y \log x} \]

7. **Multiply by \(x(1 - y \log x \log y)\)**:
Now, we need to find \( x(1 - y \log x \log y) \frac{dy}{dx} \):
\[ x(1 - y \log x \log y) \frac{dy}{dx} = x(1 - y \log x \log y) \cdot \frac{\log a \cdot y x^{y-1}}{\frac{1}{y} - \log a \cdot x^y \log x} \]

### Final Expression:
Thus, the final expression for \( x(1 - y \log x \log y) \frac{dy}{dx} \) is:
\[ x(1 - y \log x \log y) \cdot \frac{\log a \cdot y x^{y-1}}{\frac{1}{y} - \log a \cdot x^y \log x} \]