Question

How do you use implicit differentiation to find dy/dx given `x^3+3x^2y+y^3=8`?

Answer 1

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

Explanation:

differentiate `color(blue)"implicitly with respect to x"`

Use the `color(blue)"product rule"" on the term " 3x^2y`

`rArr3x^2+3x^2.dy/dx+6xy+3y^2.dy/dx=0`

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

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

`=(-cancel(3)(x^2+2xy))/(cancel(3)(x^2+y^2))`

`=-(x^2+2xy)/(x^2+y^2)`