Let ` y = x^(x^(x...oo))," then " (dy)/(dx) ` is equal to

Options

A. `yx^(y-1)`
B. `(y^(2))/(x(1-ylnx))`
C. `(y)/(x(1+ylnx))`
D. None of these

Answer

To solve the problem, we start with the equation given:

Let \( y = x^{(x^{(x^{...})})} \).

This implies that:

\[ y = x^y \]

### Step 1: Taking the logarithm of both sides

Taking the natural logarithm on both sides gives us:

\[ \log y = \log(x^y) \]

Using the property of logarithms, we can simplify the right-hand side:

\[ \log y = y \log x \]

### Step 2: Differentiating both sides with respect to \( x \)

Now, we differentiate both sides with respect to \( x \):

Using implicit differentiation on the left side:

\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(y \log x) \]

Using the product rule on the right side:

\[ \frac{d}{dx}(y \log x) = \frac{dy}{dx} \log x + y \cdot \frac{1}{x} \]

### Step 3: Setting up the equation

Now we have:

\[ \frac{1}{y} \frac{dy}{dx} = \frac{dy}{dx} \log x + \frac{y}{x} \]

### Step 4: Rearranging the equation

Rearranging gives us:

\[ \frac{1}{y} \frac{dy}{dx} - \frac{dy}{dx} \log x = \frac{y}{x} \]

Factoring out \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} \left(\frac{1}{y} - \log x\right) = \frac{y}{x} \]

### Step 5: Solving for \( \frac{dy}{dx} \)

Now, we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{y}{x} \cdot \frac{1}{\frac{1}{y} - \log x} \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{y^2}{x(1 - y \log x)} \]

### Final Result

Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{y^2}{x(1 - y \log x)} \]