If `y=a^x^a^x^(dot)dot^((((oo)))),p rov et h a t(dy)/(dx)=(y^2logy)/(x(1-ylogxdotlogy)`

Explanation

To solve the problem and prove that \(\frac{dy}{dx} = \frac{y^2 \log y}{x(1 - y \log x \log y)}\), we will follow these steps:

### Step 1: Define the function
Let \( y = a^{x^{a^{x^{\cdots}}}} \). This means that \( y = a^{x^y} \) because the exponent itself is \( y \).

**Hint:** Recognize that the infinite exponentiation can be simplified by substituting \( y \) back into the equation.

### Step 2: Take the logarithm of both sides
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