what is the derivative of x to the power of the natural log of x

Question

It looks like...

Find the derivative of the function

y = x^ln x

Answer 1

The derivative of x^ln(x).

x=e^ln(x) so x^ln(x)=(e^ln(x))^ln(x)=e^ln²(x).

(d/dx)(e^ln²(x))=

e^ln²(x)(d/dx)(ln²(x))=

e^ln²(x)2ln(x)(1/x)=

2ln(x)e^ln²(x)/x=2ln(x)x^ln(x)/x.

Let's break this down into steps. Let u=ln(x), then du/dx=1/x.

If y=x^ln(x), then, since x=e^ln(x), y=(e^ln(x))^ln(x)=(e^u)^u=e^u².

Let v=u², then dv/du=2u, and y=e^v.

dy/du=(dy/dv)(dv/du)=e^v(2u)=2ue^u².

dy/dx=(dy/du)(du/dx)=(2ue^u²)/x=2ln(x)y/x=2ln(x)x^ln(x)/x.