Answer
$(a)$. $$\frac{dy}{dx}=\frac{20x^3}{\cos y}$$
$(b)$. The slope of the tangent line to the curve is $m=-20$
Work Step by Step
$(a)$ Use implicit differentiation to determine $\frac{dy}{dx}$ for $\sin y=5x^4-5 $
Taking the derivative implicitly we get:
$$\cos y\frac{dy}{dx}=20x^3$$
solve for $\frac{dy}{dx}$
$$\frac{dy}{dx}=\frac{20x^3}{\cos y}$$
$(b)$. Find the slope of the tangent line to the curve at $(1, \pi)$
We plug in the point $(1,\pi)$ into the derivative from part $(a)$. Hence:
$$\frac{dy}{dx}=\frac{20(1)^3}{\cos (\pi)}=\frac{20}{-1}=-20$$
