# How do you use the Quotient Rule to differentiate the function f (t) = (cos (7t)) / t^5?

Jul 15, 2015

I found: $f ' \left(t\right) = - \frac{7 t \sin \left(7 t\right) + 5 \cos \left(7 t\right)}{t} ^ 6$

#### Explanation:

The Quotient Rule can be written as:
Given $f \left(t\right) = \frac{h \left(t\right)}{g \left(t\right)}$ then: $\textcolor{red}{f ' \left(t\right) = \frac{h ' \left(t\right) g \left(t\right) - h \left(t\right) g ' \left(t\right)}{g \left(t\right)} ^ 2}$

$f ' \left(t\right) = \frac{- 7 \sin \left(7 t\right) \cdot {t}^{5} - 5 {t}^{4} \cos \left(7 t\right)}{{t}^{5}} ^ 2 =$
$= \frac{- {t}^{4} \left(7 t \sin \left(7 t\right) + 5 \cos \left(7 t\right)\right)}{t} ^ 10 =$
$= - \frac{7 t \sin \left(7 t\right) + 5 \cos \left(7 t\right)}{t} ^ 6$