The Derivative of x ^x

^{When taking the derivative of} x ^{x it often looks very hard but this is one of the functions you can always understand if you can understand how it is derived.}

^y=x x

^{The trick here is to take the ln of both sides which allows us to use the logarithmic power rule.}

lny=lnx ^x

^{Power rule of logs:}

lny=xlnx

Take the derivative of both sides and on the right side we will use the quotient rule and on the left side we will use implicit differentiation and the chain rule.

y '(1/y)=lnx +x(1/x)=lnx+1

y '=y(lnx+1)

We know y=x ^x  

                          ^so

^{y '=}x ^x(lnx+1)