help me find the nth derivative of x raised to the power of x ,.....i.e x^(x)
Use the logarithmic derivative:
$\displaystyle y = x^x$
$\displaystyle ln(y) = ln \left ( x^x \right )$
Now take the derivative:
$\displaystyle \dfrac{1}{y} \cdot \dfrac{dy}{dx} = ln(x) + x \cdot \dfrac{1}{x}$
$\displaystyle \dfrac{1}{y} \cdot \dfrac{dy}{dx} = ln(x) + 1$
$\displaystyle \dfrac{dy}{dx} = y ( ln(x) + 1) = ( ln(x) + 1) x^x$
-Dan