Note(2) stated is not correct.

if y=`sqrt(ax)`

`y'=a/(2sqrt(ax))`

`y=sqrt(1+xe^(-2x))`

`y'=(1/2)*((1+xe^(-2x))^((1/2)-1)) *d/dx (sqrt(1+xe^(-2x)))`

`y'=(1/(2sqrt(1+xe^(-2x)))) *(xd/dx e^(-2x) +e^(-2x)d/dx x)`

`y'=(1/(2sqrt(1+xe^(-2x)))) *(xe^(-2x)*(-2) + e^(-2x))`

`y'=(e^(-2x) *(1-2x))/(2sqrt(1+xe^(-2x)))`

**Note:- 1) if y = e^(ax) ; then dy/dx = a*e^(ax)**

**2) if y = sqrt(ax) ; then dy/dx = [1/sqrt(ax)]*a**

Thus,

y = sqrt[1 + {x*e^(-2x)}]

dy/dx = y' = [1/2 sqrt[1 + {x*e^(-2x)}]]*[e^(-2x) - {2*x*e^(-2x)}]

or, dy/dx = y' = [(e^-2x)*(1-2x)]/2 sqrt[1 + {x*e^(-2x)}]

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