Question

suppose that x and y are related by the given equation and use implicit differentiatio

x²·y²=2

dy/dx=□

Explanation:

Step1: Differentiate both sides

Differentiate \(x^{2}y^{2}\) using the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = x^{2}\), \(u^\prime=2x\), \(v = y^{2}\), \(v^\prime = 2y\frac{dy}{dx}\).
Differentiating the right - hand side: \(\frac{d}{dx}(2)=0\).
So, \(\frac{d}{dx}(x^{2}y^{2})=\frac{d}{dx}(2)\) gives \(2xy^{2}+2x^{2}y\frac{dy}{dx}=0\).

Step2: Solve for \(\frac{dy}{dx}\)

First, factor out \(2xy\) from the left - hand side: \(2xy(y + x\frac{dy}{dx})=0\).
Since \(x\) and \(y\) are related by \(x^{2}y^{2}=2\) (so \(xy
eq0\)), we can divide both sides by \(2xy\).
We get \(y + x\frac{dy}{dx}=0\).
Then, isolate \(\frac{dy}{dx}\): \(x\frac{dy}{dx}=-y\), so \(\frac{dy}{dx}=-\frac{y}{x}\).

Answer:

\(-\frac{y}{x}\)