Derivative rules (part 2)

class: center, middle, inverse, title-slide

# Derivative rules (part 2)
## Differential Calculus
### Arturo Sánchez González
### <a href="mailto:arturo.sanchez@upaep.mx" class="email">arturo.sanchez@upaep.mx</a>
### April 2021

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# Warming up

Find the derivative of the following functions.

.pull-left[
- `$f(x) = \frac{1}{x^{12}}$`

- `$g(x) = -\frac{1}{127^{12}}$` 
]

.pull-right[
- `$h(t) = \sqrt[5]{357}$`

- `$\ell(x) = \sqrt[7]{x^{8}}$` 
]

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# Multiplying by constants

Let `$c$` be a constant, and `$f(x)$` a function.

**The derivative of `$c\cdot f(x)$`** is

$$\frac{d}{dx}\left[ c\cdot f(x)\right] = c \cdot \frac{df}{dx}(x) = c\cdot f'(x)$$

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# How to find the derivative? (1/2)

Take `$g(x) = 3x^{4}$`.

## 1st step

Identify every part of the function: Which is the constant? What is the function `$f(x)$`?

.pull-left[
 **Constant**: `$c =3$` 
]

.pull-right[
 **Function**: `$f(x) = x^{4}$` 
]

## 2nd step

"Pull out" the constant when deriving.

`$$\frac{dg}{dx}(x) = \frac{d}{dx} \left[3\cdot f(x)\right] = 3 \frac{d}{dx}\left[ f(x) \right]$$`

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# How to find the derivative? (2/2)

## 3rd step

Find the derivative of the function `$f(x)$`.

`$$\frac{dg}{dx}(x) = 3 \frac{d}{dx}\left[ f(x)\right] = 3 \left(4x^{4-1}\right)$$`

## 4th step

Make all the necessary simplifications.

`$$\frac{dg}{dx}(x) = 12x^{3}$$`

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# Examples

.pull-left[

- `$f(x) = 10x^{6}$`

- `$g(t) = -4t^{5}$`
 ]

.pull-right[

- `$h(x) = -\frac{2}{x^{3}}$`

- `$\ell(t) = 8 \sqrt{t^{3}}$`
 ]