Question

If \(y=x^{x^{x\cdot \cdot \cdot \infty }}\), then \(\frac{dy}{dx}\) is:

• A. \(yx^{y-1}\)
• B. \(\frac{y^{2}}{x(1-y\log x)}\)
• C. \(\frac{y}{x(1+y\log x)}\)
• D. \(yx^{y+1}\)

Hint:

Notice that the tower repeats itself, so y = x^y; take logarithms and differentiate implicitly.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The exponent tower is the same as \(y\) itself, so \(y=x^{y}\).

Step 2: Take logarithm
\(\log y=y\log x\).

Step 3: Differentiate
\(\frac{1}{y}\frac{dy}{dx}=\frac{dy}{dx}\log x+\frac{y}{x}\).

Step 4: Solve for dy/dx
\(\frac{dy}{dx}\left(\frac{1}{y}-\log x\right)=\frac{y}{x}\Rightarrow \frac{dy}{dx}=\frac{y^{2}}{x(1-y\log x)}\).

Final Answer:
The derivative is \(\frac{y^{2}}{x(1-y\log x)}\). \[ \boxed{\text{(B) }\dfrac{y^{2}}{x(1-y\log x)}} \]