Question

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

Answer 1

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