implicit differentiation
take the deritivive with respect to x
first
y+2=(1/2)(x^2)-2sin(y)
take deritivitive
dy/dx+0=1/2(2x)-2(cosy)dy/dx
dy/dx=x-2cosy(dy/dx)
add 2cosy(dy/dx) to both sides
dy/dx+2cosy(dy/dx) =x
undistribute dy/dx
dy/dx(1+2cos(y))=x
divide both sides by 1+2cos(y)
\(\frac{dy}{dx}= \frac{x}{1+2cos(y)}\)
at (2,0)
x=2
y=0
\(\frac{dy}{dx}= \frac{2}{1+2cos(0)}\)
\(\frac{dy}{dx}= \frac{2}{1+2(1)}\)
\(\frac{dy}{dx}= \frac{2}{1+2}\)
\(\frac{dy}{dx}= \frac{2}{3}\)
the slope is 2/3
