I am learning about implicit differentiation. The author gave 2 examples of differentiation as shown below:
Example 1: $$\frac{d}{dx}[x^3]=3x^2$$
Example 2: $$\frac{d}{dx}[y^3]=3y^2\frac{dy}{dx}$$
The explanation for adding$$\frac{dy}{dx}$$ since y is a function of x.
But where is the "y" in this equation? The only y I see is $$y^3$$
In fact, $$\frac{d(y^3)}{dx}=\frac{d(y^3)}{\color{blue}{dy}}\frac{\color{blue}{dy}}{dx}=3y^2\frac{dy}{dx}$$
Example: A circle $x^2+y^2=9$. If you want $\frac{dy}{dx}$, you'll have to write $y$ as subject of the equation. But with implicit differentiation: $$\frac{d}{dx}\left(x^2+y^2\right)=\frac{d}{dx}(9)$$ and $$2x+\frac{d}{dx}(y^2)=0=2x+\frac{d}{dy}(y^2)\frac{dy}{dx}$$