question let y be defined implicitly by the equation (-8x - 2y)^3=-x^2 - 10y^2. use implicit differentiation…

Explanation:

Step1: Differentiate both sides

Differentiate $(-8x - 2y)^3=-x^2 - 10y^2$ with respect to $x$. Using the chain - rule on the left - hand side and the power rule on both sides. The derivative of the left - hand side: $3(-8x - 2y)^2(-8 - 2\frac{dy}{dx})$. The derivative of the right - hand side: $-2x-20y\frac{dy}{dx}$. So, $3(-8x - 2y)^2(-8 - 2\frac{dy}{dx})=-2x - 20y\frac{dy}{dx}$.

Step2: Expand the left - hand side

Expand $3(-8x - 2y)^2(-8 - 2\frac{dy}{dx})$: [ \begin{align*} 3(-8x - 2y)^2(-8)-3(-8x - 2y)^2(2\frac{dy}{dx})&=-2x - 20y\frac{dy}{dx}\ -24(-8x - 2y)^2-6(-8x - 2y)^2\frac{dy}{dx}&=-2x - 20y\frac{dy}{dx} \end{align*} ]

Step3: Isolate $\frac{dy}{dx}$ terms

Move all terms with $\frac{dy}{dx}$ to one side: [ \begin{align*} -6(-8x - 2y)^2\frac{dy}{dx}+20y\frac{dy}{dx}&=-2x + 24(-8x - 2y)^2\ \frac{dy}{dx}(20y-6(-8x - 2y)^2)&=-2x + 24(-8x - 2y)^2 \end{align*} ]

Step4: Solve for $\frac{dy}{dx}$

[ \frac{dy}{dx}=\frac{-2x + 24(-8x - 2y)^2}{20y-6(-8x - 2y)^2}=\frac{-x + 12(-8x - 2y)^2}{10y-3(-8x - 2y)^2} ]

Answer:

$\frac{-x + 12(-8x - 2y)^2}{10y-3(-8x - 2y)^2}$