Find the derivative of y = \frac {\cos x}{x^3}.

Question:

Find the derivative of {eq}y = \dfrac {\cos x}{x^3} {/eq}.

Quotient rule of differentiation:

Let us consider the function {eq}f(x) = \dfrac {u(x)}{v(x)} {/eq}, then the derivative of the function with respect to {eq}x {/eq} is given by,

{eq}f'(x) = \dfrac {u'(x) \cdot v(x) - v'(x) \cdot u(x)}{(v(x))^2} {/eq}


Power rule of differentiation:

The derivative of {eq}x^n {/eq} is {eq}nx^{n - 1} {/eq}, here {eq}n {/eq} is a real number.

Answer and Explanation:

Given :

{eq}y = \dfrac {\cos x}{x^3} {/eq}

Differentiate the function with respect to {eq}x {/eq},

{eq}\displaystyle \begin{align*} y' &= \dfrac {d}{dx} \left [ \dfrac {\cos x}{x^3} \right ] \\ &= \dfrac { - \sin x \cdot x^3 - 3x^2 \cdot \cos x}{\left ( x^3 \right )^2} \\ &= - \dfrac {x^3\sin x + 3x^2 \cos x}{x^6} \\ &= - \dfrac {x^2\left ( x \ \sin x + 3\cos x \right )}{x^6} \\ y' &= - \dfrac { x \ \sin x + 3\cos x}{x^4} \\ \end{align*} {/eq}


Learn more about this topic:

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Quotient Rule: Formula & Examples

from Division: Help & Review

Chapter 1 / Lesson 5
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