Product rule

class: center, middle, inverse, title-slide

# Product rule
## 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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# Warming up

Find the derivatives of the following functions.

.pull-left[
- `$f(t) = 56 -5 t$`

- `$g(x) = 2\ln (x) -\frac{2}{3}\cos (x)$`
]

.pull-right[
- `$h(t) = -5\sin (t) - \frac{4}{5}e^{t}$`

- `$\ell (x) = \frac{2}{5}x^{6} - 16 + 4\ln(x)$`
]

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# Product rule

</center>
(3Blue1Brown, 2017)

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# Product rule

If `$f(x)$` `$=$` `$g(x)$` `$\cdot$` `$h(x)$` , then

<center>

 `$\frac{df}{dx}(x)$` `$=$` `$g(x)$` `$\cdot$` `$\frac{dh}{dx}(x)$` `$+$` `$\frac{dg}{dx}(x)$` `$\cdot$` `$h(x)$`

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# Examples (1/2)

Find the derivative of the following function

<center>

`$f(x) =$` `$x^{5}$` `$\cos(x)$`

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# Examples (2/2)

Find the derivative of the following function

<center>

`$s(t) =$` `$e^{t}$` `$\ln(t)$`

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# Dig deeper

Find the derivative of the following function

`$$h(x) = 5\cos (x) + x^{4}\ln (x) +2 - e^{x}\cos(x)$$`

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# References

- 3Blue1Brown. (2017, May 01). Visualizing the chain rule and product rule | eoc #4. Retrieved May 12, 2021, from https://www.youtube.com/watch?v=YG15m2VwSjA&amp;list=PL0-GT3co4r2wlh6UHTUeQsrf3mlS2lk6x&amp;index=4