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

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (x^{2}y-2x^{2}-4 = 0) term - by - term. Using the product rule ((uv)^\prime=u^\prime v + uv^\prime) (where (u = x^{2}) and (v=y)), the derivative of (x^{2}y) is (2xy+x^{2}\frac{dy}{dx}). The derivative of (-2x^{2}) is (-4x), and the derivative of (-4) is (0). So, (\frac{d}{dx}(x^{2}y)-\frac{d}{dx}(2x^{2})-\frac{d}{dx}(4)=\frac{d}{dx}(0)) gives (2xy + x^{2}\frac{dy}{dx}-4x=0).

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

Isolate the terms with (\frac{dy}{dx}): (x^{2}\frac{dy}{dx}=4x - 2xy). Then (\frac{dy}{dx}=\frac{4x - 2xy}{x^{2}}=\frac{4 - 2y}{x}) (divide numerator and denominator by (x), assuming (x\neq0)).

Step3: Find the slope at the point ((2,3))

Substitute (x = 2) and (y = 3) into (\frac{dy}{dx}=\frac{4 - 2y}{x}). (\frac{dy}{dx}\big|_{(x = 2,y = 3)}=\frac{4-2\times3}{2}=\frac{4 - 6}{2}=\frac{-2}{2}=-1).

Answer:

(\frac{dy}{dx}=\frac{4 - 2y}{x}), and the slope at the point ((2,3)) is (-1).