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

## I calculated this and got the right answer according to Symbolab. My solution was $\left({x}^{{x}^{2}}\right) \left(x + 2 x \ln x\right)$However, the solution for $y = {x}^{{x}^{2}}$ on the worksheet I've been provided is $\frac{\mathrm{dy}}{\mathrm{dx}} = \left({x}^{\left({x}^{2}\right) + 1}\right) \left(2 \ln x + 1\right)$ How have they managed to get this?

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

$\ln y = {x}^{2} \ln x$

Differentiate

$\frac{y '}{y} = 2 x \ln x + x$

Multiply by $y$

$y ' = {x}^{{x}^{2}} \cdot \left(2 x \ln x + x\right)$

Take $x$ as a common factor

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

$y ' = x . {x}^{{x}^{2}} \cdot \left(2 \ln x + 1\right)$

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

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

$y ' = {x}^{{x}^{2} + 1} \cdot \left(2 \ln x + 1\right)$