question let y be defined implicitly by the equation 4x^{2}+5y^{2}=-9xy. use implicit differentiation to…

Explanation:

Step1: Differentiate both sides

Differentiate $4x^{2}+5y^{2}=- 9xy$ with respect to $x$. Using the sum - rule, product - rule and power - rule: The derivative of $4x^{2}$ with respect to $x$ is $8x$ (by power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$). The derivative of $5y^{2}$ with respect to $x$ is $10y\frac{dy}{dx}$ (by chain - rule $\frac{d}{dx}(f(y))=f^{\prime}(y)\frac{dy}{dx}$). The derivative of $-9xy$ with respect to $x$ is $-9y-9x\frac{dy}{dx}$ (by product - rule $\frac{d}{dx}(uv)=u^{\prime}v + uv^{\prime}$, where $u=-9x$ and $v = y$). So, $8x + 10y\frac{dy}{dx}=-9y-9x\frac{dy}{dx}$.

Step2: Solve for $\frac{dy}{dx}$

Move all terms with $\frac{dy}{dx}$ to one side: $10y\frac{dy}{dx}+9x\frac{dy}{dx}=-9y - 8x$. Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(10y + 9x)=-9y - 8x$. Then $\frac{dy}{dx}=\frac{-9y - 8x}{9x + 10y}$.

Step3: Evaluate at the point $(2,-2)$

Substitute $x = 2$ and $y=-2$ into $\frac{dy}{dx}$: $\frac{dy}{dx}\big|_{(2,-2)}=\frac{-9(-2)-8(2)}{9(2)+10(-2)}=\frac{18 - 16}{18-20}=\frac{2}{-2}=-1$.

Answer:

$-1$