Question

16 multiple choice 1 point differentiate implicitly to find the slope of the curve at the given point. $x^2 + y^2 + 2y = 0$; $(0, -2)$ 2 1 0 -2

Explanation:

Step1: Implicit Differentiation

Differentiate each term of \(x^{2}+y^{2}+2y = 0\) with respect to \(x\).
For \(x^{2}\), derivative is \(2x\).
For \(y^{2}\), use chain rule: \(2y\frac{dy}{dx}\).
For \(2y\), derivative is \(2\frac{dy}{dx}\).
For \(0\), derivative is \(0\).
So, \(2x + 2y\frac{dy}{dx}+2\frac{dy}{dx}=0\).

Step2: Solve for \(\frac{dy}{dx}\)

Factor out \(\frac{dy}{dx}\):
\(2x+\frac{dy}{dx}(2y + 2)=0\)
Subtract \(2x\):
\(\frac{dy}{dx}(2y + 2)=-2x\)
Divide by \(2y + 2\):
\(\frac{dy}{dx}=\frac{-2x}{2y + 2}=\frac{-x}{y + 1}\).

Step3: Substitute Point \((0, -2)\)

Plug \(x = 0\), \(y=-2\) into \(\frac{dy}{dx}\):
\(\frac{dy}{dx}=\frac{-0}{-2 + 1}=\frac{0}{-1}=0\).

Answer:

\(0\) (corresponding to the option with value \(0\))