Derivative of $y=x^{\ln x}$?

Question

I only know how to do one step:

$$ \ln\left(\,y\,\right) = \ln\left(\, x^{\ln\left(\, x\,\right)}\,\right) $$ how do i do the derivative of $\ln\left(\, x^{\ln\left(\, x\,\right)}\,\right)$ ?. I know the answer to that is $\ln\left(\,x\,\right)\ln\left(\,x\,\right)$, but how does the $x$ go away ?.

Answer 1

Hint: $x^{\ln x} = e^{(\ln x)^2}$

Answer 2

We have $$\ln(y) = \ln^2(x) \implies \dfrac{dy}y = \dfrac{2\ln(x)dx}x \implies \dfrac{dy}{dx} = \dfrac{2y\ln(x)}x = 2x^{\ln(x)-1}\ln(x)$$

Answer 3

You have $$ y(x) = x^{\log x} = \exp (\log x \log x) = \exp(\log^2 x) $$ Now using the chain rule: $$ y'(x) = \exp(\log^2 x) \times 2\frac{\log x} x = 2x^{\log x - 1}\log x $$

Answer 4

If $$\ln{y}=\ln{x^{\ln{x}}}=\ln{x}\cdot\ln{x}=\ln^2{x}$$ then

$$\frac{y'}{y}=2\ln{x}\cdot\frac{1}{x}\implies y'=2y\frac{\ln{x}}{x}=2x^{\ln{x}}\cdot\frac{\ln{x}}{x}=2x^{(\ln{x}-1)}\ln{x}$$

Answer 5

Here are the steps $$ \frac{d}{dx}\left[x^{\ln x}\right]=\frac{d}{dx}\left[e^{\ln x^{\ln x}}\right] =\frac{d}{dx}\left[e^{(\ln x)(\ln x)}\right]=\frac{d}{dx}\left[e^{(\ln x)^2}\right]=e^{(\ln x)^2}\frac{d}{dx}\left[(\ln x)^2\right]=e^{(\ln x)^2}(2\ln x)\frac{d}{dx}\left[\ln x\right]=e^{(\ln x)^2}(2\ln x)\frac{1}{x}=\frac{x^{\ln x}}{x}(2\ln x)=2x^{\ln x - 1}\ln x $$

Answer 6

$$\Big(x^{\ln x}\Big)'~=~\Big(x^n\Big)'_{n~{\large=}~\ln x}+\Big(a^{\ln x}\Big)'_{a~{\large=}~x}~=~\Big(n\cdot x^{n-1}\Big)_{n~{\large=}~\ln x}+\bigg(a^{\ln x}\cdot\ln a\cdot\ln'x\bigg)_{a~{\large=}~x}~=~$$

$$~=~\ln x\cdot x^{\ln x-1}+x^{\ln x}\cdot\dfrac{\ln x}x~=~2\cdot x^{\ln x-1}\cdot\ln x.$$