1.If xy = 4 ,prove that x(dy/dx + y2) =3y..2.If y=(ex - e -x/ex +e-x ),dy/dx=1-y2.3.Ify=(x-1) log(x-1)-(x+1)log(x+1),prove that dy/dx=log(x-1/1+x).

1. y=4/x, thus dy/dx= -4/x², x(dy/dx + y²) =x(-4/x² + 16/x²) = 12/x=3y

2.from quotient rule, dy/dx = ((e^x+e^-x)(e^x+e^-x)-(e^x-e^-x)(e^x-e^-x))/(e^x+e^-x)²= 1-y²

3) from product rule, dy/dx= (x-1)/(x-1) + log(x-1) - (x+1)/(x+1) -log(x+1) = log((x-1)/(x+1))

Approved