$f^{\prime }(x)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$

The derivative $f^{\prime }(a)$ is the instantaneous rate of change of $y=f(x)$ with respect to $x$ when $x=a$:
$v(a)=\underset{h\to 0}{lim}{\displaystyle \frac{f(a+h)-f(a)}{h}}=f^{\prime }(a)$

1. a corner
2. a discontinuity
3. a vertical tangent

$\frac{d}{dx}[cf(x)]=c\frac{d}{dx}f(x)$

$\frac{d}{dx}[f(x)g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

$\frac{d}{dx}c=0$

$$\begin{gathered} {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{dy}{du}}{\displaystyle \frac{du}{dx}} \\ \frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x) \end{gathered}$$

$dy=f^{\prime }(x)dx$

$\underset{\theta \to 0}{lim}{\displaystyle \frac{\sin \theta }{\theta }}=1$

$\underset{h\to 0}{lim}{\displaystyle \frac{e^h-1}{h}}=1$

$e=\underset{n\to \text{infin}}{lim}(1+\frac{1}{n}{)}^n$

$0$

$n{x}^{n-1}$

$\cos x$

$-\sin x$

${\displaystyle \frac{1}{1+x^2}}$