Question

question
let y be defined implicitly by the equation
(2x + 9y)^2=-8x^2 - 10y^2.
use implicit differentiation to find $\frac{dy}{dx}$.
provide your answer below:
$\frac{dy}{dx}=square$

Explanation:

Step1: Expand left - hand side

Using the formula $(a + b)^2=a^{2}+2ab + b^{2}$, we expand $(2x + 9y)^{2}$ to get $4x^{2}+36xy + 81y^{2}$. So the equation becomes $4x^{2}+36xy + 81y^{2}=-8x^{2}-10y^{2}$.

Step2: Rearrange the equation

Move all terms to one side: $4x^{2}+36xy + 81y^{2}+8x^{2}+10y^{2}=0$, which simplifies to $12x^{2}+36xy + 91y^{2}=0$.

Step3: Differentiate term - by - term

Differentiate $12x^{2}$ with respect to $x$: $\frac{d}{dx}(12x^{2}) = 24x$.
Differentiate $36xy$ with respect to $x$ using the product rule $(uv)^\prime=u^\prime v+uv^\prime$ where $u = 36x$ and $v = y$. So $\frac{d}{dx}(36xy)=36y + 36x\frac{dy}{dx}$.
Differentiate $91y^{2}$ with respect to $x$ using the chain rule. Let $u = y$, then $\frac{d}{dx}(91y^{2})=182y\frac{dy}{dx}$.
The derivative of the left - hand side of the equation is $24x+36y + 36x\frac{dy}{dx}+182y\frac{dy}{dx}$, and the derivative of the right - hand side (0) is 0.

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

Group the terms with $\frac{dy}{dx}$: $36x\frac{dy}{dx}+182y\frac{dy}{dx}=-24x - 36y$.
Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(36x + 182y)=-24x - 36y$.
Then $\frac{dy}{dx}=\frac{-24x - 36y}{36x+182y}=\frac{-12x - 18y}{18x + 91y}$.

Answer:

$\frac{-12x - 18y}{18x + 91y}$