Quotient Rule for Derivatives/Examples

Examples of Use of Quotient Rule for Derivatives

Example: $\dfrac x {x + 1}$

$\map {\dfrac \d {\d x} } {\dfrac x {x + 1} } = \dfrac 1 {\paren {x + 1}^2}$

Example: $\dfrac {x + 1} {x - 1}$

$\map {\dfrac \d {\d x} } {\dfrac {x + 1} {x - 1} } = -\dfrac 2 {\paren {x - 1}^2}$

Example: $\dfrac x {\cos x}$

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

Example: $\dfrac {e^x} x$

$\map {\dfrac \d {\d x} } {\dfrac {e^x} x} = \dfrac {e^x \paren {x - 1} } {x^2}$

Example: $\dfrac {\paren {x - 1} \paren {2 x - 1} } {x - 2}$

$\map {\dfrac \d {\d x} } {\dfrac {\paren {x - 1} \paren {2 x - 1} } {x - 2} } = \dfrac {2 x^2 - 8 x + 5} {\paren {x - 2}^2}$