# Formulas of derivative

In fact, I don't wanna write this. But I'm in china school, they don't care what you have learned, they only care about what you can do on a test paper.

## How to read

$$\underline{sine} = \sin$$

$$\underline{co} \underline{sine} = \cos$$

$$\underline{tan} \underline{gent} = \tan$$

$$\underline{co} \underline{tangent} = \cot$$

$$\underline{se} \underline{cant} = \sec$$

$$\underline{cose} \underline{cant} = \csc$$

## Basic Derivative Formula

$$
\begin{align*}
\\

(C)^\prime &= 0    &\text{(C is a constant)} \\ \\

(x^n)^\prime &= nx^{n-1} \\ \\

(a^x)^\prime &= a^x\ln{a}    &(e^x)^\prime &= e^x \\ \\

(\log_a{x})^\prime &= \frac{1}{x\ln{a}}    &(\ln{x})^\prime &= \frac{1}{x} \\ \\ \\


(\sin{x})^\prime &= \cos{x}    &(\cos{x})^\prime &= -\sin{x} \\ \\

(\tan{x})^\prime &= \sec^2{x}    &(\cot{x})^\prime &= -\csc^2{x} \\ \\

(\sec{x})^\prime &= \sec{x}\tan{x}    &(\csc{x})^\prime &= -\csc{x}\cot{x} \\ \\ \\


(\arcsin{x})^\prime &= \frac{1}{\sqrt{1 - x^2}}    &(\arccos{x})^\prime &= -\frac{1}{\sqrt{1 - x^2}} \\ \\

(\arctan{x})^\prime &= \frac{1}{1 + x^2}    &(arccot{x})^\prime &= -\frac{1}{1 + x^2} \\ \\

\end{align*}
$$

## Additional

$$
\begin{align*}
&\csc x = \frac{1}{\sin x}
\\ \\
&\sec x = \frac{1}{\cos x}
\\ \\
&\cot x = \frac{1}{\tan x}
\\ \\ \\
&\sin ^2{x} + \cos ^2{x} = 1
\\ \\
&1 + \tan ^2{x} = \sec ^2{x}
\\ \\
&1 + \cot ^2{x} = \csc ^2{x}
\\ \\ \\
&\tan{x} = \frac{\sin{x}}{\cos{x}}
\\ \\
&\cot{x} = \frac{\cos{x}}{\sin{x}}
\end{align*}
$$

