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)#However, the solution for #y=x^(x^2)# on the worksheet I've been provided is #dy/dx = (x^((x^2)+1))(2lnx+1)#
How have they managed to get this?

1 Answer
Yahia M.
Apr 24, 2018

check the explanation below

Explanation:

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

#y=x^(x^2)#

Taking the natural logarithm of both sides

#lny=x^2lnx#

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)#

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