Question

What is the implicit derivative of `1= ysqrt(xy)-y`?

Answer 1

`dy/dx=-(ysqrt(xy))/(x(3sqrt(xy)-2))`

Explanation:

`d/dx[1=ysqrt(xy)-y]`

`0=sqrt(xy)dy/dx+yd/dx[sqrt(xy)]-dy/dx`

`0=sqrt(xy)dy/dx+y(1/(2sqrt(xy)))d/dx[xy]-dy/dx`

`0=sqrt(xy)dy/dx+y/(2sqrt(xy))(y+xdy/dx)-dy/dx`

`0=sqrt(xy)dy/dx+y^2/(2sqrt(xy))+(xy)/(2sqrt(xy))(dy/dx)-dy/dx`

`0=sqrt(xy)dy/dx+(ysqrt(xy))/(2x)+(sqrt(xy))/2(dy/dx)-dy/dx`

`-(ysqrt(xy))/(2x)=dy/dx(sqrt(xy)+(sqrt(xy))/2-1)`

`-(ysqrt(xy))/(2x)=dy/dx((3sqrt(xy)-2)/2)`

`dy/dx=-(ysqrt(xy))/(2x)(2/(3sqrt(xy)-2))`

`dy/dx=-(ysqrt(xy))/(x(3sqrt(xy)-2))`

Please ask with any questions! Above all, never forget to use the product rule.