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# Use derivative formula

## 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 $$dy$$.

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