Quotient Rule for Derivatives/Examples/(x-1)(2x-1) over x-2
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Example of Use of Quotient Rule for Derivatives
- $\map {\dfrac \d {\d x} } {\dfrac {\paren {x - 1} \paren {2 x - 1} } {x - 2} } = \dfrac {2 x^2 - 8 x + 5} {\paren {x - 2}^2}$
Proof
| \(\ds \map {\dfrac \d {\d x} } {\dfrac {\paren {x - 1} \paren {2 x - 1} } {x - 2} }\) | \(=\) | \(\ds \dfrac {\paren {x - 2} \map {\frac \d {\d x} } {\paren {x - 1} \paren {2 x - 1} } - \paren {x - 1} \paren {2 x - 1} \map {\frac \d {\d x} } {x - 2} } {\paren {x - 2}^2}\) | Quotient Rule for Derivatives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {x - 2} \paren {\paren {2 x - 1} \map {\frac \d {\d x} } {x - 1} + \paren {x - 1} \map {\frac \d {\d x} } {2 x - 1} } - \paren {x - 1} \paren {2 x - 1} \cdot 1} {\paren {x - 2}^2}\) | Product Rule for Derivatives, Derivative of Identity Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {x - 2} \paren {\paren {2 x - 1} \cdot 1 + \paren {x - 1} \cdot 2} - \paren {x - 1} \paren {2 x - 1} } {\paren {x - 2}^2}\) | Derivative of Identity Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {x - 2} \paren {2 x - 1} + 2 \paren {x - 1} \paren {x - 2} - \paren {x - 1} \paren {2 x - 1} } {\paren {x - 2}^2}\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 x^2 - 4 x - x + 2 + 2 x^2 - 2 x - 4 x + 4 - 2 x^2 + 2 x + x - 1} {\paren {x - 2}^2}\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 x^2 - 8 x + 5} {\paren {x - 2}^2}\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $28$.