Find the slope of the graph of $xy-2y^2=8$ at $(10,4)$
So there are two different routes we could take:
We could implicitly differentiate, then solve for $\frac{dy}{dx}$ (the slope), then plug in the point $(10,4)$.
Or, we could implicitly differentiate, then plug in the point $(10,4)$, then solve for $\frac{dy}{dx}$ (the slope).
in my solution I will solve for $\frac{dy}{dx}$ and THEN plug in the point $(10,4)$
Solution:
$$\frac{d}{dx}(xy-2y^2)=\frac{d}{dx}8=0$$
$$\left(\frac{d}{dx}(xy)-2\frac{d}{dx}(y^2)\right)=0$$
We will need to use the product rule to evaluate $\frac{d}{dx}(xy)$:
$$\left(\left(\frac{d}{dx}x\right)y+\left(x\left(\frac{d}{dx}y\right)\right)-2\left(2y\frac{dy}{dx}\right)\right)=0$$
$$(1)y+\left(x(1)\frac{dy}{dx}\right)-4y\frac{dy}{dx}=0$$
$$y+x\frac{dy}{dx}-4y\frac{dy}{dx}=0$$
$$(x-4y)\frac{dy}{dx}=-y$$
$$\frac{dy}{dx}=\frac{-y}{x-4y}$$
Cool, so we have now took the implicit derivative and then solved for $\frac{dy}{dx}$, now we plug in $(10,4)$
$$\frac{dy}{dx}=\frac{-4}{10-4(4)}$$
$$\frac{dy}{dx}=\frac{-4}{-6}$$
$$\frac{dy}{dx}=\frac{2}{3}$$
And so the slope of $xy-2y^2=8$ at $(10,4)$ is $\frac{2}{3}$
Your solution is correct.
In general, it is important to remember to perform implicit differentiation when you are attempting to find the slope at a point in which it is difficult to separate $y$.