Implicit equation - $G(x,y)=0$

Question

I'm confused about some points in implicit equation ...

From my recitation class -

$G(x,y)=0$ provides - $y=f(x)$ .

And $f'(x)=\frac{dy}{dx}$

and about $G'(x,y)=0$ we use -

How would be look $f'(x)$ in case of the follow implicit equation ? -

Answer 1

I have usually seen this called an implicit function. You are not guaranteed the ability to transform it into the form $y=f(x)$, but you can do implicit differentiation. From $3x^7+2y^5-x^3+y^3-3=0$ you take a derivative with respect to $x$ to get $21x^6+10y^4y'-3x^2+3y^2y'=0, y'=-\frac{21x^6-3x^2}{10y^4+3y^2}$

Answer 2

You have

$$G(x,y)=3x^7+2y^5-x^3+y^3-3=0 $$

so

$$\frac{\partial G}{\partial x}=21x^6-3x^2 $$

$$\frac{\partial G}{\partial y}=10y^4+3y^2 $$

therefore

$$\frac{dy}{dx}=-\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial y}}=-\frac{21x^6-3x^2}{10y^4+3y^2} $$