Implicit Differentiation

Example 1:

$$ x^2+y^2=4, \;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ x^2+y^2=4$$ $$ \frac{d}{dx}x^2+\frac{d}{dx}y^2=\frac{d}{dx}4$$ $$ \frac{d}{dx}x^2+\frac{d}{dy}y^2 \frac{dy}{dx}=\frac{d}{dx}4$$ $$ \because\;\; \frac{d}{dx} x^n=nx^{n-1} \;\;\;\textup{and} \;\;\; \frac{d}{dx}c=0$$ $$ 2x^{2-1}+2y^{2-1}\frac{dy}{dx}=0$$ $$ 2x+2y\frac{dy}{dx}=0$$ $$ 2y\frac{dy}{dx}=-2x$$ $$ \frac{dy}{dx}=\frac{-2x}{2y}$$ $$ \frac{dy}{dx}=-\frac{x}{y}$$