Which of the following statements are true?

I. $\frac{d}{dx}\left(y^2\right) = 2y\frac{dy}{dx}$.

II. $\frac{d}{dx}(xy) = x\frac{dy}{dx} + y$.

III. $\frac{d}{dx}(\sin y) = \cos y$.

IV. For $x^2 + y^2 = 16$, $\frac{dy}{dx} = -\frac{x}{y}$.

Explanation

1. 01

   Every function of $y$ picks up a factor $\frac{dy}{dx}$ when differentiated with respect to $x$: that makes I true and III false, since $\frac{d}{dx}(\sin y) = \cos y\frac{dy}{dx}$.

2. 02

   II is the product rule. IV: $2x + 2y\frac{dy}{dx} = 0$ gives $-\frac{x}{y}$, so it is true.