Implicit differentiation and inverse trigonometric functions practice question 16

The sine is applied to $\pi y$, and $y$ depends on $x$. What is the derivative of that inside with respect to $x$?

Differentiate each term with respect to $x$. By the chain rule, $\dfrac{d}{dx}\sin(\pi y) = \cos(\pi y)\cdot\pi\,\dfrac{dy}{dx}$, because the inside $\pi y$ has derivative $\pi\,\dfrac{dy}{dx}$.

Collect the two $\dfrac{dy}{dx}$ terms and factor. At $y = 1$ the angle is $\pi$, and $\cos\pi = -1$.