Could you please differentiate with respect to x and simplify? `x^(x^2 )` thanks

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lemjay | High School Teacher | (Level 3) Senior Educator

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`y=x^(x^2)`

Before differentiating with respect to x, express this as a logarithmic equation. To do so, take the logarithm of both sides.

`lny=lnx^(x^2)`

At the right side, apply the property `lna^m=mlna` .

`lny=x^2lnx`

Then, differentiate both sides with respect to x.

`d/(dx)lny=d/(dx)(x^2 lnx)`

At the left side, apply the rule `(lnu)'=1/u*u'` .

`1/y* (dy)/(dx)=d/(dx)(x^2 lnx)`

And at the right side apply, the product rule `(u*v)=u*v'+v*u'` .

`1/y* (dy)/(dx)=x^2(lnx)'+lnx(x^2)`

`1/y* (dy)/(dx)=x^2*1/x+lnx*(2x)`

`1/y* (dy)/(dx)=x+2xlnx`

`1/y* (dy)/(dx)=x(1+2lnx)`

Then, isolate dy/dx.

`dy/dx=x(1+2lnx)*y`

Since `y=x^(x^2)` , plug-in this to dy/dx.

`dy/dx=x(1+2lnx)*x^(x^2)`

`dy/dx=x^(x^2+1)(1+2lnx)`

Hence, `dy/dx=x^(x^2+1)(1+2lnx)` .

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