(a) differentiate the given equation implicitly and then solve for y. (b) solve the given equation for y and…

Explanation:

Step1: Differentiate $2x^{2}$

Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, for $a = 2$ and $n = 2$, we have $\frac{d}{dx}(2x^{2})=4x$.

Step2: Differentiate $-8y$

Using the chain - rule, $\frac{d}{dx}(-8y)=-8y'$.

Step3: Differentiate $-4$

The derivative of a constant is 0, so $\frac{d}{dx}(-4) = 0$.

Step4: Set up the differentiated equation

Differentiating $2x^{2}-8y - 4=0$ term - by - term gives $4x-8y'-0 = 0$.

Step5: Solve for $y'$

First, isolate the term with $y'$: $8y'=4x$. Then divide both sides by 8 to get $y'=\frac{4x}{8}=\frac{x}{2}$.

Answer:

$y'=\frac{x}{2}$