Using logarithmic differentiation or otherwise, differentiate x^(x^2)?

I calculated this and got the right answer according to Symbolab. My solution was (x^(x^2))(x+2xlnx)(xx2)(x+2xlnx)However, the solution for y=x^(x^2)y=xx2 on the worksheet I've been provided is dy/dx = (x^((x^2)+1))(2lnx+1)dydx=(x(x2)+1)(2lnx+1)
How have they managed to get this?

1 Answer
Apr 24, 2018

check the explanation below

Explanation:

color(blue)("The two answers are the same just difference in simplification"The two answers are the same just difference in simplification

y=x^(x^2)y=xx2

Taking the natural logarithm of both sides

lny=x^2lnxlny=x2lnx

Differentiate

(y')/y=2xlnx+x

Multiply by y

y'=x^(x^2)*(2xlnx+x)

Take x as a common factor

color(green)("This is the part You forgot to simplify"

y'=x.x^(x^2)*(2lnx+1)

color(green)(x^axxx^b=x^(a+b)

color(green)(x^1xxx^(x^2)=x^(x^2+1)

y'=x^(x^2+1)*(2lnx+1)