Chapter 3 - Differentiation - 3.8 Implicit Differentiation - Exercises - Page 152: 2

$$ y+x\frac{dy}{dx}+4+2\frac{dy}{dx}=0.$$ $$\frac{dy}{dx}=-\frac{y+4}{x+2}.$$ $$\frac{dy}{dx}|_{(1,-1)}=-\frac{3}{3}=-1.$$

Differentiating the equation $ xy+4x+2y=1$ with respect to $ x $ and using the chain rule, we get $$ y+x\frac{dy}{dx}+4+2\frac{dy}{dx}=0.$$ Hence, $$\frac{dy}{dx}=-\frac{y+4}{x+2}.$$ Moreover, we have $$\frac{dy}{dx}|_{(1,-1)}=-\frac{3}{3}=-1.$$