# Complete the following derivative \frac{d}{dx} \left ( \frac {\cos{x}}{x^2} \right )

## Question:

Complete the following derivative {eq}\frac{d}{dx} \left ( \frac {\cos{x}}{x^2} \right ) {/eq}

## Quotient Rule:

Let {eq}g(x) {/eq} and {eq}m(x) {/eq} be functions of {eq}x {/eq} then the derivative of {eq}\frac{{g(x)}}{{m(x)}} {/eq} is given by:

{eq}{\,\frac{d}{{dx}}\frac{g}{m} = \frac{{m\frac{{dg}}{{dx}} - g\frac{{dm}}{{dx}}}}{{{m^2}}}} {/eq}

The following rules are relevant to this problem:

1.{eq}{\frac{d}{{dx}}{x^n} = n{x^{n - 1}}} {/eq}

2.{eq}{\frac{d}{{dx}}\cos x = - \sin x} {/eq}

Given that: {eq}\displaystyle \frac{d}{{dx}}\left( {\frac{{\cos x}}{{{x^2}}}} \right) {/eq}

{eq}\displaystyle \eqalign{ & \frac{d}{{dx}}\left( {\frac{{\cos x}}{{{x^2}}}} \right) \cr & {\text{Using derivative properties;}} \cr & = \frac{{{x^2}\frac{d}{{dx}}\cos x - \cos x\frac{d}{{dx}}{x^2}}}{{{x^4}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{From quotient rule}}} \right) \cr & {\text{From known derivative;}} \cr & = \frac{{{x^2}\left( { - \sin x} \right) - \cos x\left( {2x} \right)}}{{{x^4}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\frac{d}{{dx}}{x^n} = n{x^{n - 1}},\frac{d}{{dx}}\cos x = - \sin x} \right) \cr & = - \frac{{x\sin x + 2\cos x}}{{{x^3}}} \cr & \cr & \frac{d}{{dx}}\left( {\frac{{\cos x}}{{{x^2}}}} \right) = - \frac{{x\sin x + 2\cos x}}{{{x^3}}} \cr} {/eq}