Answer
$(a)$ $$\frac{dy}{dx}=-\csc y$$
$(b)$ The slope of the tangent line to the curve is $m=-1$
Work Step by Step
$(a)$ Use implicit differentiation to determine $\frac{dy}{dx}$ for $\cos y=x$
Taking the derivative implicitly we get:
$$-\sin y\frac{dy}{dx}=1$$
solve for $\frac{dy}{dx}$
$$\frac{dy}{dx}=\frac{1}{-\sin y}=-\csc y$$
$(b)$ Find the slope of the tangent line to the curve at $(0,\pi/2 )$
We plug in the point $(0,\pi/2 )$ into the derivative from part $(a)$. Hence:
$$\frac{dy}{dx}=-\csc (\pi/2)=-1$$
