logarithmic differentiation

[vickig]

how do you differentiate y=the square root of x raised to the x power?

 

[pka]

how do you differentiate y=the square root of x raised to the x power?

First: Let \(\displaystyle y=\sqrt{x^x}\) then \(\displaystyle \log(y)=\dfrac{x}{2}\log(x)\)

Now the derivative: \(\displaystyle \dfrac{y'}{y}=\dfrac{1}{2}\log(x)+\dfrac{x}{2} \cdot\dfrac{1}{x}\).

Solve for \(\displaystyle y'\).