What is the derivative of #y=x^lnx#?

1 Answer
Dec 16, 2015

#(2x^(lnx)lnx)/x#

Explanation:

Take natural log of both sides

#lny=lnx^(lnx)#

Rewrite write hand side using properties of logarithms

#lny=lnx(lnx)#

#lny=(lnx)^2#

Differentiate both sides with respect to #x#

#1/y(dy)/dx=2lnx(1/x)#

#1/y(dy)/dx=(2lnx)/x#

Multiply both sides by #y#

#(dy)/dx=y((2lnx)/x)#

Note that #y=x^(lnx)# so we write

#(dy)/dx=x^(lnx)((2lnx)/x)#

#(dy)/dx=(2x^(lnx)lnx)/x#