Identify the base and exponent in $x^{x^x}$ in order to apply power rule of differentiation

Question

While differentiating ${x}^{{x}^{x}}$ using power rule, what should be the base and exponent, i.e. base=$x$, exponent=$x^x$ or base=$x^x$, exponent=$x$. Any WHY?

Answer 1

Think of $x^{x^x}$ as $$f(x)^{g(x)}\tag 1$$ where $f(x)=x$ and $g(x)=x^x$. I you consider $$F(x)^{G(x)}\tag 2$$ with $F(x)=x^x$ and $G(x)=x$ then you will have $(x^x)^x=x^{x^2}$ which is different from $(1)$.

So, stick to $(1)$. There is a generalized power rule to be applied to $(1)$: $$\frac d{dx}f(x)^{g(x)}=g(x)f(x)^{g(x)-1}\frac {df}{dx}+f(x)^{g(x)}\ln(f(x))\frac {dg}{dx}$$

if $f(x)>0$. We have $\frac{df}{dx}=1$ and $\frac{dg}{dx}=x^x(1+\ln(x)).$ The final result can be given now. Note that this is true if $f(x)=x>0$.