Question

What is the implicit derivative of `y=x^3y+x^2y^4-5y`?

Answer 1

`dy/dx =- (2x^2y + 3xy^2)/(x^3 + 2x^2y - 6)`

Explanation:

Let's first put all our variables on one side.

`x^3y + x^2y^2 - 5y - y = 0`

By a combination of implicit differentiation and the product rule, we can differentiate without having to isolate `y`.

`3x^2y + x^3(dy/dx) + 2xy^2 + 2yx^2(dy/dx) - 6(dy/dx) = 0`

`x^3(dy/dx) + 2x^2y(dy/dx) - 6(dy/dx) = -3x^2y - 2xy^2`

`dy/dx(x^3 + 2x^2y - 6) = -3x^2y - 2xy^2`

`dy/dx =- (2x^2y + 3xy^2)/(x^3 + 2x^2y - 6)`

Hopefully this helps!