Quotient rule

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# Quotient 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.

$$f(x) = \left( x^4 - 3\ln(x)\right)\cos (x)$$

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# Quotient rule

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

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

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# Examples (1/2)

Find the derivative of the following function.

$$f(x) = \frac{e^{x}}{x^2}$$

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# Examples (2/2)

Find the derivative of the following function.

$$g(x) = \tan (x)$$

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# Let's practice! (1/2)

Find the derivative of the following function.

$$s(t) = \frac{3t^2}{\ln (t)}$$

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# Let's practice! (2/2)

Find the derivative of the following function.

$$r(x) = \frac{3x^5 - \cos (x)}{e^{x}}$$