differentiate implicitly to find \\( \\frac { d y } { d x } \\). then, find the slope of the curve at the…

differentiate implicitly to find \\( \\frac { d y } { d x } \\). then, find the slope of the curve at the given point\n\\( 4 x ^ { 2 } - 4 y ^ { 2 } = 8, \\quad ( \\sqrt { 5 } , \\sqrt { 3 } ) \\)\n\\( \\frac { d y } { d x } = \\square \\)\nthe slope of the curve at \\( ( \\sqrt { 5 } , \\sqrt { 3 } ) \\) is \\( \\square \\).\n(type an exact answer, using radicals as needed )

differentiate implicitly to find \\( \\frac { d y } { d x } \\). then, find the slope of the curve at the given point\n\\( 4 x ^ { 2 } - 4 y ^ { 2 } = 8, \\quad ( \\sqrt { 5 } , \\sqrt { 3 } ) \\)\n\\( \\frac { d y } { d x } = \\square \\)\nthe slope of the curve at \\( ( \\sqrt { 5 } , \\sqrt { 3 } ) \\) is \\( \\square \\).\n(type an exact answer, using radicals as needed )

Answer

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (4x^{2}-4y^{2}=8). Using the power rule ((x^{n})^\prime = nx^{n - 1}) and the chain - rule ((y^{n})^\prime=ny^{n - 1}\frac{dy}{dx}). Differentiating (4x^{2}) gives (8x). Differentiating (-4y^{2}) gives (-8y\frac{dy}{dx}). Differentiating (8) gives (0). So, (8x-8y\frac{dy}{dx}=0).

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

Rearrange (8x - 8y\frac{dy}{dx}=0) to isolate (\frac{dy}{dx}). First, move (8x) to the other side: (-8y\frac{dy}{dx}=-8x). Then divide both sides by (-8y) (assuming (y\neq0)), we get (\frac{dy}{dx}=\frac{x}{y}).

Step3: Find the slope at the point ((\sqrt{5},\sqrt{3}))

Substitute (x = \sqrt{5}) and (y=\sqrt{3}) into (\frac{dy}{dx}=\frac{x}{y}). So, (\frac{dy}{dx}\big|_{x = \sqrt{5},y=\sqrt{3}}=\frac{\sqrt{5}}{\sqrt{3}}=\frac{\sqrt{15}}{3}).

Answer:

(\frac{dy}{dx}=\frac{x}{y}); The slope of the curve at ((\sqrt{5},\sqrt{3})) is (\frac{\sqrt{15}}{3})