Find the Derivative of f(x) = x^log(x) using Logarithmic Differentiation

[Fairy111]

Homework Statement

Find the derivative of the following function f.

Homework Equations

f(x)=x^log(x)

The Attempt at a Solution

I really don't know how to do this, as it's x to the power of a function of x.

 

[D H]

Use the fact that for any function u(x),

$$u(x) = e^{\ln(u(x))}$$

 

[HallsofIvy]

Or (almost the same thing) use "logarithmic differentiation".

If y= x^{log x} then log(y)= log(x^log(x))= log(x)log(x).

Differentiate both sides with respect to x, using the chain rule on the left side, and then solve for dy/dx.