Question

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?

Answer 1

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)`