suppose that x and y are related by the equation x² - 4y² = 3 and use implicit differentiation to determine…

Explanation:

Step1: Differentiate both sides

Differentiate $x^{2}-4y^{2}=3$ with respect to $x$. The derivative of $x^{2}$ with respect to $x$ is $2x$ by the power - rule. For $-4y^{2}$, we use the chain - rule. Let $u = y$, then $\frac{d(-4y^{2})}{dx}=-8y\frac{dy}{dx}$. The derivative of the constant 3 with respect to $x$ is 0. So we have $2x-8y\frac{dy}{dx}=0$.

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

Isolate $\frac{dy}{dx}$ in the equation $2x - 8y\frac{dy}{dx}=0$. First, move $2x$ to the other side: $-8y\frac{dy}{dx}=-2x$. Then divide both sides by $-8y$ (assuming $y\neq0$). We get $\frac{dy}{dx}=\frac{x}{4y}$.

Answer:

$\frac{x}{4y}$