Question

for the following exercises, use implicit differentiation to find \\( \frac{d y}{d x} \\).

1. \\( x^{2}-y^{2}=4 \\)
2. \\( 6 x^{2}+3 y^{2}=12 \\)

Explanation:

Step1: Differentiate both sides of the equation

Differentiate \(x^{2}-y^{2}=4\) with respect to \(x\).
Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\) and the chain rule \(\frac{d}{dx}(y^{n})=ny^{n-1}\frac{dy}{dx}\).
We get \(2x-2y\frac{dy}{dx}=0\).

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

Rearrange the equation \(2x-2y\frac{dy}{dx}=0\) to isolate \(\frac{dy}{dx}\).
First, move \(2x\) to the other side: \(- 2y\frac{dy}{dx}=-2x\).
Then divide both sides by \(-2y\) (assuming \(y
eq0\)): \(\frac{dy}{dx}=\frac{x}{y}\).

Step1: Differentiate both sides of the equation

Differentiate \(6x^{2}+3y^{2}=12\) with respect to \(x\).
Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\) and the chain rule \(\frac{d}{dx}(y^{n})=ny^{n-1}\frac{dy}{dx}\).
We have \(12x + 6y\frac{dy}{dx}=0\).

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

Rearrange the equation \(12x+6y\frac{dy}{dx}=0\) to isolate \(\frac{dy}{dx}\).
First, move \(12x\) to the other side: \(6y\frac{dy}{dx}=-12x\).
Then divide both sides by \(6y\) (assuming \(y
eq0\)): \(\frac{dy}{dx}=-\frac{2x}{y}\).

Answer:

\(\frac{dy}{dx}=\frac{x}{y}\)

---