Question

question let y be defined implicitly by the equation (-8x - 2y)^3=-x^2 - 10y^2. use implicit differentiation to find dy/dx. provide your answer below: dy/dx =

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:

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