Quotient Rule

Reference > Calculus: Differentiation

Description
(fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}
Examples
ddxsin⁡xx2\frac{d}{dx} \frac{\sin{x}}{{x}^{2}}
1
Use Quotient Rule to find the derivative of sin⁡xx2\frac{\sin{x}}{{x}^{2}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
x2(ddxsin⁡x)−sin⁡x(ddxx2)x4\frac{{x}^{2}(\frac{d}{dx} \sin{x})-\sin{x}(\frac{d}{dx} {x}^{2})}{{x}^{4}}

2
Use Trigonometric Differentiation: the derivative of sin⁡x\sin{x} is cos⁡x\cos{x}.
x2cos⁡x−sin⁡x(ddxx2)x4\frac{{x}^{2}\cos{x}-\sin{x}(\frac{d}{dx} {x}^{2})}{{x}^{4}}

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
x2cos⁡x−2xsin⁡xx4\frac{{x}^{2}\cos{x}-2x\sin{x}}{{x}^{4}}

Done

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