What is the slope of the line tangent to the curve y+2=(x^2/2)-2sin(y) at the point (2,0)?

dy/dx = x - 2cos(y) dy/dx
dy/dx(1 + 2cos(y)) = x
dy/dx = x/(1 + 2cos(y))

at (2,0), m = dy/dx = 2/(1 + 2cos(0)) = 2/(1 + 2) = 2/3

Derive implicitly

y + 2 = (1/2) * x^2 - 2 * sin(y)
dy + 0 = (1/2) * 2x * dx - 2 * cos(y) * dy
dy = x * dx - 2 * cos(y) * dy

x = 2 , y = 0

dy = 2 * dx - 2 * cos(0) * dy
dy = 2 * dx - 2 * 1 * dy
dy = 2 * dx - 2 * dy
dy + 2 * dy = 2 * dx
3 * dy = 2 * dx
3 * dy/dx = 2
dy/dx = 2/3

y + 2 = (x²/2)-2siny at point (2, 0)

dy/dx + 0 = 2x/2 - 2cosy.dy/dx

dy/dx = x-2cosy dy/dx

dy/dx + 2cosy.dy/dx = x

dy/dx (1+2cosy) = x

dy/dx = (x/1 + 2cosy) / (2, 0)

= 2/1 + 2cos0 = 2/3.

take a differential
dy = 2xdx / 2 - 2 cos(y) dy
dy / dx = x / ( 1 + 2cos(y))
dy / dx = 2 / ( 1 + 2cos(0))
dy / dx = 2/3