Calculate derivative of x^2+xy+2y^2 = 4 with implicit differentiation?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to use the implicit differentiation to evaluate the derivative of the given function, hence, you need to consider `y` a function of `x` , such that:

`(d(x^2+xy+2y^2))/(dx) = (d(4))/(dx)`

`2x + (d(xy))/(dx) + 4y*(dy)/(dx) = 0`

Use the product rule for the middle term, such that:

`2x + y + x*(dy)/(dx) + 4y*(dy)/(dx) = 0`

You need to isolate the terms that contain `(dy)/(dx)` to the left side, such that:

`x*(dy)/(dx) + 4y*(dy)/(dx) = -2x - y`

Factoring out `(dy)/(dx)` yields:

`(dy)/(dx)*(x + 4y) = -(2x + y)`

`(dy)/(dx) = -(2x + y)/(x + 4y)`

Hence, evaluating the derivative of the given function using implicit differentiation, yields `(dy)/(dx) = -(2x + y)/(x + 4y).`