1. ## implicit differentiation

Can anyone show me how to find $\frac{dy}{dx}$ for $(x^2+y^2-1)(y-x^2-1) = 0$ at the points (0,1) and (0,0).

I know that I need to use the process of implicit differentiation but I don't know exactly how to apply it here. Do I need to expand the brackets first and then differentiate everything?

2. Originally Posted by demode
Can anyone show me how to find $\frac{dy}{dx}$ for $(x^2+y^2-1)(y-x^2-1) = 0$ at the points (0,1) and (0,0).

I know that I need to use the process of implicit differentiation but I don't know exactly how to apply it here. Do I need to expand the brackets first and then differentiate everything?
you don't have to expand the brackets. just use the product rule. remember, when differentiating a y-term, you need to attach dy/dx to it.

3. JHevon brings up a good point, but if you DO expand the brackets, several terms cancel, thus making it easier to calculate the derivative.

$(x^2 + y^2 - 1)(y - x^2 - 1) = 0$

$x^2(y - x^2- 1) + y^2(y - x^2 - 1) - 1(y - x^2 - 1) = 0$

$x^2y - x^4 - x^2 + y^3 - x^2y - y^2 - y + x^2 + 1 = 0$

$-x^4 + y^3 - y^2 - y + 1 = 0$

$y^3 - y^2 - y + 1 = x^4$

$\frac{d}{dx}(y^3 - y^2 - y + 1) = \frac{d}{dx}(x^4)$

$\frac{d}{dy}(y^3 - y^2 - y + 1)\frac{dy}{dx} = 4x^3$

$(3y^2 - 2y - 1)\frac{dy}{dx} = 4x^3$

$(3y + 1)(y - 1)\frac{dy}{dx} = 4x^3$

$\frac{dy}{dx} = \frac{4x^3}{(3y + 1)(y - 1)}$.