Question

If $x=y^{x^{y^x.......\infty }}\text{then}\frac{dy}{dx}\text{at}x=1$ is

• A. 1
• B. 2
• C. 5
• D. 7

Answer 1

The correct option is A 1

$x=y^{x^{y^x.......\infty }}\Rightarrow x=y^{x^x}\Rightarrow \ln x=x^x\ln y$
Differentiating on both sides $\Rightarrow \frac{1}{x}=x^x(1+\ln x)\ln y+\frac{x^x}{y}.y^{\prime }$
Also $x=1\Rightarrow y=1$
Therefore $y^{\prime }=1$