Chain rule

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# Chain 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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# Quick recall

Find the derivative of the following function.

`$$\color{blue}{v(t)} = \frac{5}{6x^{5}}$$`

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# Chain rule

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

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

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# Examples (1/2)

Find the derivative of the following function.

`$$\color{green}{f(x)} = \color{red}{\sin}\left( \color{blue}{x^2}\right)$$`

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# Examples (2/2)

Find the derivative of the following function.

`$$v(t) = e^{t^2 - 5}$$`

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# Let's practice

Find the derivative of the following function.

`$$\color{darkblue}{\ell(t) = \left( \ln(t) \right)^2 }$$`

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# Derivative rules and practice

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(3Blue1Brown, 2017)

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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