in exercises 17 - 24, find dy/dx by implicit differentiation and evaluate the derivative at the indicated…

Explanation:

Step1: Differentiate both sides

Differentiate $xy$ with respect to $x$ using product - rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = x$ and $v = y$. The derivative of the left - hand side is $y + x\frac{dy}{dx}$, and the derivative of the right - hand side (a constant 4) is 0. So we have $y + x\frac{dy}{dx}=0$.

Step2: Solve for $\frac{dy}{dx}$

Isolate $\frac{dy}{dx}$: [ \begin{align*} x\frac{dy}{dx}&=-y\ \frac{dy}{dx}&=-\frac{y}{x} \end{align*} ]

Step3: Evaluate at the given point

Substitute $x=-4$ and $y = - 1$ into $\frac{dy}{dx}=-\frac{y}{x}$: [ \frac{dy}{dx}\big|_{x = - 4,y=-1}=-\frac{-1}{-4}=-\frac{1}{4} ]

Answer:

$-\frac{1}{4}$