Question

Differentiate the following functions $w.r.t.x$:
$x^{x^x}$

Answer 1

We have,

${\left(x^x\right)}^x$

Let,

$y={\left(x^x\right)}^x=x^{x^x}$

Taking log both side and we get,

$\mathrm{logy}=log\left(x^{x^x}\right)$

$\log y=x^x\log x$

Again taking log both side and we get,

$\log \log y=\log \left(x^x\log x\right)$

$\Rightarrow \log \log y=\log x^x+\log \log x$

$\Rightarrow \log \log y=x\log x+\log \log x$

On differentiation and we get,

$\frac{1}{\log y}\frac{1}{y}\frac{dy}{dx}=x\frac{d}{dx}\log x+\log x\frac{dx}{dx}+\frac{1}{x\log x}$

$\Rightarrow \frac{1}{y\log y}\frac{dy}{dx}=x\times \frac{1}{x}+\log x+\frac{1}{x\log x}$

$\Rightarrow \frac{dy}{dx}=y\log y(1+\log x+\frac{1}{x\log x})$

$\Rightarrow \frac{dy}{dx}=x^{x^x}\log x^x(1+\log x+\frac{1}{x\log x})$

Hence, this is the answer.