Quotient Rule for Derivatives/Examples/Sine of x over x

Example of Use of Quotient Rule for Derivatives

$\map {\dfrac \d {\d x} } {\dfrac {\sin x} x} = \dfrac {x \cos x - \sin x} {x^2}$

Proof

\(\ds \map {\dfrac \d {\d x} } {\dfrac {\sin x} x}\)

\(=\)

\(\ds \dfrac {x \map {\frac \d {\d x} } {\sin x} - \sin x \map {\frac \d {\d x} } x} {x^2}\)

Quotient Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \dfrac {x \cdot \cos x - \sin x \cdot 1} {x^2}\)

Derivative of Sine Function, Power Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \dfrac {x \cos x - \sin x} {x^2}\)

simplification

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quotient rule
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quotient rule