Question

How do you differentiate `x^3-e^(x-xy)-4y^2=2xy`?

Answer 1

`dy/dx=(3x^2-e^(x-xy)+ye^(x-xy)-2y)/(8y+2x+xe(x-xy)`

Explanation:

Since it is not possible to write `y` as an explicit function of `x`, we use the method of implicit differentiation.

This entails differentiating both sides of the equation with respect to `x`, bearing in mind that `y` is a function of `x`, and then solving for `dy/dx`.

Doing so yields :

`d/dx(x^3-e^(x-xy)-4y^2)=d/dx(2xy)`

`therefore 3x^2-[(e^(x-xy))*(1-xdy/dx-y)]-8ydy/dx=2xdy/dx+2y`

`thereforedy/dx=(3x^2-e^(x-xy)+ye^(x-xy)-2y)/(8y+2x+xe(x-xy)`