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) [tex] \frac{dy}{dx}= \frac{x}{1+2cos(y)} [/tex] at (2,0) x=2 y=0
[tex] \frac{dy}{dx}= \frac{2}{1+2cos(0)} [/tex] [tex] \frac{dy}{dx}= \frac{2}{1+2(1)} [/tex] [tex] \frac{dy}{dx}= \frac{2}{1+2} [/tex] [tex] \frac{dy}{dx}= \frac{2}{3} [/tex] the slope is 2/3
