A quick review of the basic rules of the derivation of functions

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# A quick review of the basic rules of the derivation of functions
## Differential Calculus
### Arturo Sánchez González
### <a href="mailto:arturo.sanchez@upaep.mx" class="email">arturo.sanchez@upaep.mx</a>
### May 2021

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# Basic rules of derivation (1/2)

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## Constant rule

If `$c$` is a fixed number, and `$f(x) = c$`, then
`$$\frac{df}{dx}(x) = f'(x) = 0$$`

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## Power rule

If `$f(x) = x^{n}$`, then
`$$\frac{df}{dx}(x) = f'(x) = nx^{n-1}$$`
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## Scalar rule
If `$c$` is a fixed number, `$f(x)$` is a function and `$g(x) = c\cdot f(x)$`, then
`$$\frac{dg}{dx}(x) = c\cdot \frac{df}{dx}(x) = c\cdot f'(x)$$`
 
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## Sum rule
If `$f(x)$` and `$g(x)$` are functions, then
`$$\frac{d}{dx}\left[ f(x) + g(x)\right] = f'(x) + g'(x)$$`
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# Basic rules of derivation (2/2)

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## Derivative of trigonometric functions

If `$f(x)=\sin (x)$`, then
`$$\frac{df}{dx}(x) = f'(x) = \cos (x)$$`

If `$g(x) = \cos(x)$`, then
`$$\frac{dg}{dx}(x) = g'(x) = -\sin (x)$$`
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## Derivative of exponential function

If `$f(x) = e^{x} = \exp(x)$`, then
`$$\frac{df}{dx}(x) = f'(x) = e^{x}$$`

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## Derivative of natural logarithm

If `$f(x) = \ln (x)$`, then
`$$\frac{df}{dx}(x) = f'(x) = \frac{1}{x}$$`
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# Calling the derivative

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# More complicated functions (1/2)

Find the derivative of the following functions.

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 `$f(x) = 3x^{5} - 4\cos(x) + 8$`
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 `$g(t) = -6 \cos (t) + \frac{5}{7} \ln(t)$`
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# More complicated functions (2/2)

Find the derivative of the following function:

`$s(x) = \frac{5}{4}x^{8} - 6\sin (x) -3\cos (x) + 10 e^{x} - 17 \ln (x)$`

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# Stretching out

Find the derivatives of the following functions.

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- `$r(x) = -45^{9}$`

- `$m(t) = \frac{2}{t^{5}} - \frac{5}{9} + 2t^{8}$`
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- `$g(t) = 4e^{t} - \frac{3}{4}\cos (t)$`

- `$\ell (x) = \frac{2}{3}\ln (x) - 9\sin (x)$`
]