Question

let y(x² + 100)=200 (witch of agnesi). complete parts a through d
a. y = -\frac{1}{10}(x - 10)-1 and y = \frac{1}{10}(x + 10)-1
b. y = -\frac{1}{10}(x - 10)-1 and y = \frac{1}{10}(x + 10)+1
c. y = -\frac{1}{10}(x)+1
d. y = -\frac{1}{10}(x - 10)+1 and y = \frac{1}{10}(x + 10)+1
c. solve the equation y(x² + 100)=200 for y to find an explicit expression for y and then calculate \frac{dy}{dx}
\frac{dy}{dx}=□

Explanation:

Step1: Solve for y

First, isolate y in the equation $y(x^{2}+100)=200$. Divide both sides by $x^{2}+100$ to get $y = \frac{200}{x^{2}+100}$.

Step2: Differentiate using quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 200$, so $u^\prime=0$, and $v=x^{2}+100$, so $v^\prime = 2x$.
Substitute into the quotient - rule:
\[

$$\begin{align*} \frac{dy}{dx}&=\frac{0\times(x^{2}+100)-200\times(2x)}{(x^{2}+100)^{2}}\\ &=\frac{- 400x}{(x^{2}+100)^{2}} \end{align*}$$

\]

Answer:

$\frac{-400x}{(x^{2}+100)^{2}}$