for the following exercises, use implicit differentiation to find \\( \\frac{d y}{d x} \\).\n300. \\(…

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\neq0)): (\frac{dy}{dx}=\frac{x}{y}).

Answer:

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

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Explanation:

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\neq0)): (\frac{dy}{dx}=-\frac{2x}{y}).

Answer:

(\frac{dy}{dx}=-\frac{2x}{y})