suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx…

Explanation:

Step1: Differentiate both sides

Differentiate $x^{4}+y^{4}=2$ with respect to $x$. The derivative of $x^{4}$ with respect to $x$ is $4x^{3}$ by the power - rule. For $y^{4}$, using the chain - rule, we get $4y^{3}\frac{dy}{dx}$. The derivative of the constant 2 is 0. So, $4x^{3}+4y^{3}\frac{dy}{dx}=0$.

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

First, isolate the term with $\frac{dy}{dx}$: $4y^{3}\frac{dy}{dx}=-4x^{3}$. Then divide both sides by $4y^{3}$ (assuming $y\neq0$) to get $\frac{dy}{dx}=-\frac{x^{3}}{y^{3}}$.

Answer:

$-\frac{x^{3}}{y^{3}}$