Question

What is the implicit derivative of `1=x/y-xtany`?

Answer 1

`dy/dx=-(y(ytany-1))/(x(1+y^2sec^2y))`

Explanation:

`d/dx[1=x/y-xtany]`

Find each derivative.

`d/dx[1]=0`

`d/dx[x/y]=(yd/dx[x]-xd/dx[y])/y^2=(y-xdy/dx)/y^2`

`d/dx[xtany]=tanyd/dx[x]+xd/dx[tany]=tany+xsec^2ydy/dx`

Plug the derivatives back in.

`0=(y-xdy/dx)/y^2-tany-xsec^2ydy/dx`

`tany=(y-xdy/dx)/y^2-xsec^2ydy/dx`

`y^2tany=y-xdy/dx-xy^2sec^2ydy/dx`

`y^2tany-y=dy/dx(-x-xy^2sec^2y)`

`dy/dx=-(y(ytany-1))/(x(1+y^2sec^2y))`