# Differentiation Rules

```{contents}
:local:
```

## General Formulas
Assume $u$ and $v$ are differentiable functions of $x$, $c$ is a constant.

$$u'=\diff{u}{x}, \quad v'=\diff{v}{x}$$

1. $\displaystyle \diff{c}{x}=0$

2. $\displaystyle \diff{}{x}(u+v)=u' + v'$

3. $\displaystyle \diff{}{x}(u-v)=u' - v'$

4. $\displaystyle \diff{}{x}(cu) = c u'$

5. $\displaystyle \diff{}{x}(uv) = u'v + u v' $

6. $\displaystyle \diff{}{x}\biggl(\frac{u}{v}\biggr) = \frac{u'v - u v'}{v^2} $

7. $\displaystyle \diff{}{x} (x^n) = n x^{n-1}$

8. $\displaystyle \diff{}{x}\biggl\{f\biggl(g(x)\biggr)\biggr\}= \diff{f}{g} \cdot \diff{g}{x}$


## Trigonometric Functions

9. $\displaystyle \diff{}{x}(\sin x) = \cos x$

10. $\displaystyle \diff{}{x}(\cos x) = -\sin x$

11. $\displaystyle \diff{}{x}(\tan x) = \sec^2 x$

12. $\displaystyle \diff{}{x}(\sec x) = \sec x \tan x$

13. $\displaystyle \diff{}{x}(\cot x) = -\csc^2 x$

14. $\displaystyle \diff{}{x}(\csc x) = -\csc x \cot x$

## Inverse Trigonometric Functions
15. $\displaystyle \diff{}{x}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$

16. $\displaystyle \diff{}{x}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$

17. $\displaystyle \diff{}{x}(\arctan x) = \frac{1}{1+x^2}$

18. $\displaystyle \diff{}{x}(\arcsec x) = \frac{1}{|x|\sqrt{x^2-1}}$

19. $\displaystyle \diff{}{x}(\arccot x) = -\frac{1}{1+x^2}$

20. $\displaystyle \diff{}{x}(\arccsc x) = -\frac{1}{|x|\sqrt{x^2-1}}$

## Exponential and Logarithmic Functions
21. $\displaystyle \diff{}{x} (e^x) = e^x$

22. $\displaystyle \diff{}{x} (a^x) = a^x \ln a$

23. $\displaystyle \diff{}{x} (\ln x) = \frac{1}{x}$

24. $\displaystyle \diff{}{x} (\log_a x) = \frac{1}{x \ln a}$

## Hyperbolic Functions

25. $\displaystyle \diff{}{x}(\sinh x) = \cosh x$

26. $\displaystyle \diff{}{x}(\cosh x) = \sinh x$

27. $\displaystyle \diff{}{x}(\tanh x) = \sech^2 ~x$

28. $\displaystyle \diff{}{x}(\sech x) = -\sech ~x \tanh x$

29. $\displaystyle \diff{}{x}(\coth x) = -\csch^2 ~x$

30. $\displaystyle \diff{}{x}(\csch x) = -\csch ~x ~\coth x$



## Inverse Hyperbolic Functions

## Parametric Equations