use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\n\\( ( 7 x y + 5 ) ^ { 2 } = 28 y…

Explanation:

Step1: Differentiate both sides

Differentiate ((7xy + 5)^2) using the chain rule ((u^2)^\prime=2u\cdot u^\prime) (where (u = 7xy+5)) and the product rule ((uv)^\prime=u^\prime v+uv^\prime) (for (u = 7x), (v = y)). Differentiate (28y) with respect to (x) using ((cy)^\prime=c\frac{dy}{dx}) ((c = 28)). $$2(7xy + 5)\left(7y+7x\frac{dy}{dx}\right)=28\frac{dy}{dx}$$

Step2: Expand the left - hand side

$$14(7xy + 5)y+14(7xy + 5)x\frac{dy}{dx}=28\frac{dy}{dx}$$ $$98xy^{2}+70y+98x^{2}y\frac{dy}{dx}+70x\frac{dy}{dx}=28\frac{dy}{dx}$$

Step3: Group the terms with (\frac{dy}{dx})

$$98x^{2}y\frac{dy}{dx}+70x\frac{dy}{dx}-28\frac{dy}{dx}=-98xy^{2}-70y$$

Step4: Factor out (\frac{dy}{dx})

$$\frac{dy}{dx}(98x^{2}y + 70x-28)=-98xy^{2}-70y$$

Step5: Solve for (\frac{dy}{dx})

$$\frac{dy}{dx}=\frac{-98xy^{2}-70y}{98x^{2}y + 70x-28}=\frac{-7y(14xy + 10)}{14(7x^{2}y+5x - 2)}=\frac{-y(14xy + 10)}{2(7x^{2}y+5x - 2)}$$

Answer:

(\frac{-y(14xy + 10)}{2(7x^{2}y+5x - 2)})