Quotient Rule for Derivatives/Examples/x+1 over x-1
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Example of Use of Quotient Rule for Derivatives
- $\map {\dfrac \d {\d x} } {\dfrac {x + 1} {x - 1} } = -\dfrac 2 {\paren {x - 1}^2}$
Proof
\(\ds \map {\dfrac \d {\d x} } {\dfrac x {x + 1} }\) | \(=\) | \(\ds \dfrac {\paren {x - 1} \map {\frac \d {\d x} } {x + 1} - \paren {x + 1} \map {\frac \d {\d x} } {x - 1} } {\paren {x - 1}^2}\) | Quotient Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {x - 1} \cdot 1 - \paren {x + 1} \cdot 1} {\paren {x - 1}^2}\) | Power Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 2 {\paren {x - 1}^2}\) | simplification |
$\blacksquare$