#### Answer

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

#### Work Step by Step

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
$$\frac{dy}{dx}|_{(1,-1)}=-\frac{3}{3}=-1.$$