## Differentiating a Super Exponential

The function
$f(x)=x^x$
us a super exponential and can be differentiated by writing
$f(x)=e^{ln(x^x}=e^{x lnx}$
and using a combination of The Chain Rule and Differentiation - The Product Rule.
Using the product rule to differentiate
$xlnx$
gives
$x \times \frac{1}{x} +1 \times lnx=1+lnx$
.
Using the chain rule to differentiate
$e^{x lnx}$
gives
$(1+lnx)e^{xlnx}=(1+lnx)x^x$
.