Question

let y(x² + 100) = 200 (witch of agnesi). complete parts a through d. a. use implicit differentiation to find $\frac{dy}{dx}$ $\frac{dy}{dx}=square$

Explanation:

Step1: Expand the left - hand side

First, expand $y(x^{2}+100)$ to get $x^{2}y + 100y=200$.

Step2: Differentiate both sides with respect to $x$

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ for $x^{2}y$ (where $u = x^{2}$, $v = y$), we have:
$\frac{d}{dx}(x^{2}y)+\frac{d}{dx}(100y)=\frac{d}{dx}(200)$.
For $\frac{d}{dx}(x^{2}y)$, we get $2xy + x^{2}\frac{dy}{dx}$, and $\frac{d}{dx}(100y)=100\frac{dy}{dx}$, and $\frac{d}{dx}(200) = 0$. So, $2xy+x^{2}\frac{dy}{dx}+100\frac{dy}{dx}=0$.

Step3: Solve for $\frac{dy}{dx}$

Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(x^{2}+100)=- 2xy$.
Then $\frac{dy}{dx}=\frac{-2xy}{x^{2}+100}$.

Answer:

$\frac{-2xy}{x^{2}+100}$