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\\( x ^ { 2 } y - 3 x ^ { 3 } - 8 = 0 ; \\quad ( 2,8 ) \\)\n\\( \\frac { d y } { d x } = \\square \\)\nthe slope of the curve at \\( ( 2,8 ) \\) is \\( \\square \\). (simplify your answer.)

differentiate implicitly to find \\( \\frac { d y } { d x } \\). then find the slope of the curve at the given point\n\\( x ^ { 2 } y - 3 x ^ { 3 } - 8 = 0 ; \\quad ( 2,8 ) \\)\n\\( \\frac { d y } { d x } = \\square \\)\nthe slope of the curve at \\( ( 2,8 ) \\) is \\( \\square \\). (simplify your answer.)

Answer

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (x^{2}y-3x^{3}-8 = 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 (-3x^{3}) is (-9x^{2}), and the derivative of (-8) is (0). So, (\frac{d}{dx}(x^{2}y-3x^{3}-8)=\frac{d}{dx}(0)) gives (2xy + x^{2}\frac{dy}{dx}-9x^{2}=0).

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

Isolate the terms with (\frac{dy}{dx}): (x^{2}\frac{dy}{dx}=9x^{2}-2xy). Then (\frac{dy}{dx}=\frac{9x^{2}-2xy}{x^{2}}=\frac{9x - 2y}{x}) ((x\neq0)).

Step3: Find the slope at the point ((2,8))

Substitute (x = 2) and (y = 8) into (\frac{dy}{dx}). (\frac{dy}{dx}\mid_{x = 2,y = 8}=\frac{9\times2-2\times8}{2}). First, calculate the numerator: (9\times2-2\times8=18 - 16=2). Then (\frac{2}{2}=1).

Answer:

(\frac{dy}{dx}=\frac{9x - 2y}{x}); The slope of the curve at ((2,8)) is (1).