Questions: Differentiate implicitly to find dy/dx. Then, find the slope of the curve at the given point. 2x^2 - y^2 = 8 ; (√5, √2) dy/dx =

Solution Steps

We start with the equation given by the problem:

\[ 2x^2 - y^2 = 8 \]

Differentiating both sides with respect to \(x\) gives:

\[ \frac{d}{dx}(2x^2) - \frac{d}{dx}(y^2) = 0 \]

This results in:

\[ 4x - 2y \frac{dy}{dx} = 0 \]

Rearranging the equation to isolate \(\frac{dy}{dx}\):

\[ 2y \frac{dy}{dx} = 4x \]

Dividing both sides by \(2y\):

\[ \frac{dy}{dx} = \frac{2x}{y} \]

Now, we substitute the coordinates of the given point \((\sqrt{5}, \sqrt{2})\) into the derived formula for \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} \bigg|_{(\sqrt{5}, \sqrt{2})} = \frac{2\sqrt{5}}{\sqrt{2}} = \sqrt{10} \]

Final Answer