Question

Dy/dx of (x^1/x)[(1-lnx)/x^2]?

Answer 1

Hello,


`y=x^{ (1)/(x)}* (1-ln(x))/(x^2) \\\\ y=x^{ (1)/(x) -2}* (1-ln(x))\\\\ ln(y)= ((1)/(x) -2})*ln(x)+ln(1-ln(x))\\\\ (d(ln(x)))/(dx) = (1)/(y) * (dy)/(dx) \\ =- (1)/(x^2)*ln(x)+( (1)/(x)-2)* (1)/(x) + (1)/(1-ln(x)) *(- (1)/(x) )\\\\ (dy)/(dx)=(x^{ (1)/(x)}* (1-ln(x))/(x^2)) *((-1)/(x^2)*ln(x)+( (1)/(x)-2)* (1)/(x) + (1)/(1-ln(x)) *(-1)/(x) )\\\\ ...`

I let you simplify.