# How do you find the first and second derivatives of f(x)=cos(x)/x^8  using the quotient rule?

Nov 5, 2015

First derivative: $- \setminus \frac{x \setminus \sin \left(x\right) + 8 \setminus \cos \left(x\right)}{{x}^{9}}$
Second derivative: $\setminus \frac{7 \setminus \sin \left(x\right) - x \setminus \cos \left(x\right)}{{x}^{9}} - \setminus \frac{9 \left(\setminus \sin \left(x\right) - 8 \setminus \cos \left(x\right)\right)}{{x}^{10}}$

#### Explanation:

The quotient $\setminus \frac{f \left(x\right)}{g \left(x\right)}$ of two derivable functions $f \left(x\right)$ and $g \left(x\right)$ is derivable for all $x$ in which $g \left(x\right) \setminus \ne 0$. And its derivative is equal to:

$\frac{d}{\mathrm{dx}} \left(\setminus \frac{f \left(x\right)}{g \left(x\right)}\right) = \setminus \frac{g \left(x\right) \setminus \cdot f ' \left(x\right) - f \left(x\right) \setminus \cdot g ' \left(x\right)}{{\left(g \left(x\right)\right)}^{2}}$

So, going back to the exercise we have:

$f \left(x\right) = \setminus \sin \left(x\right)$
$g \left(x\right) = {x}^{8}$

Thus:

$\frac{d}{\mathrm{dx}} \left(\setminus \frac{f \left(x\right)}{g \left(x\right)}\right) = \setminus \frac{{x}^{8} \setminus \cdot \left(- \setminus \sin \left(x\right)\right) - \setminus \cos \left(x\right) \setminus \cdot 8 {x}^{7}}{{\left({x}^{8}\right)}^{2}}$

$\frac{d}{\mathrm{dx}} \left(\setminus \frac{f \left(x\right)}{g \left(x\right)}\right) = \setminus \frac{- {x}^{8} \setminus \sin \left(x\right) - 8 {x}^{7} \setminus \cos \left(x\right)}{{x}^{16}}$

Factoring:

$\frac{d}{\mathrm{dx}} \left(\setminus \frac{f \left(x\right)}{g \left(x\right)}\right) = \setminus \frac{- {x}^{7} \left(x \setminus \sin \left(x\right) + 8 \setminus \cos \left(x\right)\right)}{{x}^{16}}$

Reducing $\frac{{x}^{7}}{{x}^{16}}$ we finally get:

$\frac{d}{\mathrm{dx}} \left(\setminus \frac{f \left(x\right)}{g \left(x\right)}\right) = - \setminus \frac{x \setminus \sin \left(x\right) + 8 \setminus \cos \left(x\right)}{{x}^{9}}$

For the second derivative your functions $f \left(x\right)$ and $g \left(x\right)$ will be:

$f \left(x\right) = - \left(x \setminus \sin \left(x\right) + 8 \setminus \cos \left(x\right)\right)$
$g \left(x\right) = {x}^{9}$

Try doing it!