Review of all the derivative rules

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# Review of all the derivative rules
## Differential Calculus
### Arturo Sánchez González
### <a href="mailto:arturo.sanchez@upaep.mx" class="email">arturo.sanchez@upaep.mx</a>
### May 2022

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# Basic rules of derivation (1/2)

## Constant rule

If `$\color{red}{c}$` is a fixed number, and `$\color{blue}{f(x)} = \color{red}{c}$`, then

`$$\color{darkblue}{\frac{df}{dx}(x)} = \color{darkblue}{f'(x)} = \color{darkred}{0}$$`

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## Power rule

If `$\color{green}{f(x)} = \color{blue}{x}\color{red}{^{n}}$`, then

`$$\color{darkgreen}{\frac{df}{dx}(x)} = \color{darkgreen}{f'(x)} = \color{red}{n}\color{blue}{x}\color{darkred}{^{n-1}}$$`

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# Basic rules of derivation (2/2)

## Scalar multiple rule

If `$\color{red}{c}$` is a fixed number, `$\color{blue}{f(x)}$` is a function, and `$\color{green}{g(x)} = \color{red}{c}\cdot \color{blue}{f(x)}$`, then

`$$\color{darkgreen}{\frac{dg}{dx}(x)} = \color{red}{c}\cdot \color{darkblue}{\frac{df}{dx}(x)} = \color{red}{c}\cdot \color{darkblue}{f'(x)}$$`

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## Sum rule

If `$\color{green}{f(x)}$` and `$\color{magenta}{g(x)}$` are functions, then

`$$\color{darkblue}{\frac{d}{dx}\left[ f(x) \pm g(x)\right]} = \color{darkgreen}{f'(x)} \pm \color{darkmagenta}{g'(x)}$$`

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# Derivative of trigonometric functions

If `$\color{green}{f(x)}=\color{red}{\sin (x)}$`, then

`$$\color{darkgreen}{\frac{df}{dx}(x)} = \color{darkgreen}{f'(x)} = \color{darkred}{\cos (x)}$$`

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If `$\color{blue}{g(x)} = \color{magenta}{\cos(x)}$`, then

`$$\color{darkblue}{\frac{dg}{dx}(x)} = \color{darkblue}{g'(x)} = \color{darkmagenta}{-\sin (x)}$$`

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## Derivative of exponential function

If `$\color{blue}{f(x)} = \color{red}{e^{x}} = \color{red}{\exp(x)}$`, then

`$$\color{darkblue}{\frac{df}{dx}(x)} = \color{darkblue}{f'(x)} = \color{red}{e^{x}}$$`

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## Derivative of natural logarithm

If `$\color{green}{f(x)} = \color{magenta}{\ln (x)}$`, then

`$$\color{darkgreen}{\frac{df}{dx}(x)} = \color{darkgreen}{f'(x)} = \color{darkmagenta}{\frac{1}{x}}$$`

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# Product rule

If `$\color{green}{h(x)} = \color{red}{f(x)} \cdot \color{blue}{g(x)}$`, then

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`$$\color{darkgreen}{\frac{dh}{dx}(x)}= \color{darkred}{f'(x)} \cdot \color{blue}{g(x)} \color{darkmagenta}{+} \color{red}{f(x)} \cdot \color{darkblue}{g'(x)}$$`

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# Quotient rule

If `$\color{green}{h(x)} = \frac{\color{red}{f(x)}}{\color{blue}{g(x)}}$`, then

`$$\color{green}{\frac{dh}{dx}(x)} = \frac{ \color{darkred}{f'(x)} \cdot \color{blue}{g(x)} - \color{red}{f(x)} \cdot \color{darkblue}{g'(x)} }{ \left(\color{blue}{g(x)}\right)^2 }$$`

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# Chain rule

Let `$\color{red}{g(x)}$` and `$\color{blue}{f(x)}$` be two functions. Then `$\color{green}{h(x)} = \color{red}{g}\left( \color{blue}{f(x)} \right)$` is also a function and

`$$\color{darkgreen}{h'(x)} = \color{darkred}{g'}\left( \color{blue}{f(x)}\right) \cdot \color{darkblue}{f'(x)}$$`

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# Example

Find the derivative of the following function:


`$$\color{blue}{v(x) = \frac{6x^3\sin (x)}{5e^{x} - \cos (\ln(x))}}$$`