differentiate implicitly to find $\\frac{dy}{dx}$. then find the slope of the curve at the given…

differentiate implicitly to find $\\frac{dy}{dx}$. then find the slope of the curve at the given point.\n$x^{2}y - 3x^{3}-4 = 0$; $(2,7)$\n$\\frac{dy}{dx}=\\square$\nthe slope of the curve at $(2,7)$ is $\\square$. (simplify your answer.)

differentiate implicitly to find $\\frac{dy}{dx}$. then find the slope of the curve at the given point.\n$x^{2}y - 3x^{3}-4 = 0$; $(2,7)$\n$\\frac{dy}{dx}=\\square$\nthe slope of the curve at $(2,7)$ is $\\square$. (simplify your answer.)

Answer

Explanation:

Step1: Differentiate each term with respect to (x)

Differentiate (x^{2}y) using the product rule ((uv)^\prime = u^\prime v+uv^\prime) (where (u = x^{2}), (v = y)), ((x^{2}y)^\prime=2xy + x^{2}\frac{dy}{dx}). Differentiate (-3x^{3}) using the power rule ((x^{n})^\prime=nx^{n - 1}), ((-3x^{3})^\prime=-9x^{2}). Differentiate (-4) (a constant) gives (0). So, differentiating (x^{2}y-3x^{3}-4 = 0) with respect to (x) 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 ((2,7))

Substitute (x = 2) and (y = 7) into (\frac{dy}{dx}=\frac{9x - 2y}{x}). (\frac{dy}{dx}\big|_{x = 2,y = 7}=\frac{9\times2-2\times7}{2}=\frac{18 - 14}{2}).

Answer:

(\frac{dy}{dx}=\frac{9x - 2y}{x}); The slope of the curve at ((2,7)) is (2).