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) ((u = 7x), (v = y)). Differentiate (28y) with respect to (x). [ \begin{align*} 2(7xy + 5)\left(7y+7x\frac{dy}{dx}\right)&=28\frac{dy}{dx}\ \end{align*} ]

Step2: Expand the left - hand side

[ \begin{align*} 2(7xy + 5)\times7y+2(7xy + 5)\times7x\frac{dy}{dx}&=28\frac{dy}{dx}\ 14y(7xy + 5)+14x(7xy + 5)\frac{dy}{dx}&=28\frac{dy}{dx} \end{align*} ]

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

[ \begin{align*} 14x(7xy + 5)\frac{dy}{dx}-28\frac{dy}{dx}&=-14y(7xy + 5)\ \frac{dy}{dx}\left[14x(7xy + 5)-28\right]&=-14y(7xy + 5) \end{align*} ]

Step4: Solve for (\frac{dy}{dx})

[ \begin{align*} \frac{dy}{dx}&=\frac{-14y(7xy + 5)}{14x(7xy + 5)-28}\ &=\frac{-y(7xy + 5)}{x(7xy + 5)-2} \end{align*} ]

Answer:

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