Derivative of $x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$

Calculus Level 3

Let: $f(x)=x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$ where $x$ is repeated an infinite number of times. Which of the following is the correct expression of $\frac{d}{dx}f(x)$ ?

Errata. Option 2: “ $f’(x)$ has not a simple algebraic expression”.

\frac{e^x[1+f(x)^x]}{f^2(x)}

f(x)

has not a simple algebraic expression

\frac{e^{f(x)}}{x\cdot f(x)}

x^x\cdot f(x)^{f(x)}

\frac{f^2(x)}{x[1-f(x)\log(x)]}

f(x)

1 solution

Brian Charlesworth
Nov 30, 2017

Taking natural logs of both sides of the equation $\large f(x) = x^{f(x)}$ , we have that $\ln(f(x)) = f(x) \times \ln(x)$ . Differentiating implicitly with respect to $x$ then yields

$\dfrac{1}{f(x)} \times f'(x) = f'(x) \times \ln(x) + f(x) \times \dfrac{1}{x} \Longrightarrow f'(x) \left(\dfrac{1}{f(x)} - \ln(x)\right) = \dfrac{f(x)}{x} \Longrightarrow \boxed{f'(x) = \dfrac{(f(x))^{2}}{x(1 - f(x)\ln(x))}}$ .

Derivative of x x x ⋅ ⋅ ⋅ x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} x x x ⋅ ⋅ ⋅

1 solution

0 pending reports

Derivative of $x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$