For composition of functions

$$\begin{align*} y &= e^{-x} \\ \\ y^\prime &= e^{-x} \times (-x)^\prime \\ \\ &= -e^{-x} \end{align*}$$

Principle: get each function's derivative, then combined each other with $\times$.

For implicit function or relation

$$x^2 + y^2 = 0$$

$$\begin{align*} \\ &2x + 2yy^\prime = 0 \\ \\ &y^\prime = \frac{-2x}{2y} \end{align*}$$

Principle: get each side derivative, adding $y^\prime$ to every part expression when there is y included needs to get derivative.

Get differential of a function

Normally, if we say differential isolately, what we mean is $dy$.

$$\begin{align*} y &= \sin{x} + x \\ \\ dy &= \cos{x}dx + dx \\ \\ dy &= (\cos{x} + 1)dx \end{align*}$$