Find a derivative

Question

Derivate x^x.

Answer 1

(x^x)(lnx +1) is the derivative of x^x

let y = x^x

take natural logs of both sides

lny = ln(x^x) = xlnx

lny = xlnx

implicitly differentiate

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

(1/y)y' = lnx + 1

multiply both sides by y

y' = y(lnx +1)

replace y with x^x

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

(x^x)' = (x^x)(lnx +1)