`"If "log_(e)(log_(e) x-log_(e)y)=e^(x^(2_(y)))(1-log_(e)x)," then find the value of "y'(e).`

Answer

`(1+e^(e^(2)))/(e)`

Explanation

We have
`log_(e)(log_(e)x-log_(e)y)=e^(x^(2)y)(1-log_(e)x)" ...(1)"`
`"For "x =e, log_(e)(1-log_(e)y)=0.`
`therefore" "y=1`
Differentiating (1), w.r.t.x, we get
`(1)/(log_(e)x-log_(e)y).((1)/(x)-(1)/(y)y')`
`=e^(x^(2)//y)cdot(2xy+x^(2)y')(1-log_(e)x)-(1)/(x)e^(x^(2)y)`
Putting x = e and y = 1, we get
`(1)/(1-0)cdot((1)/(e)-y')=0-(1)/(e)cdote^(e^(2))`
`therefore" "y'(e)=(1+e^(e^(2)))/(e)`