Quotient Rule for Derivatives/Examples/x over x+1

Example of Use of Quotient Rule for Derivatives

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

Proof

\(\ds \map {\dfrac \d {\d x} } {\dfrac x {x + 1} }\)

\(=\)

\(\ds \dfrac {\paren {x + 1} \map {\frac \d {\d x} } x - x \map {\frac \d {\d x} } {x + 1} } {\paren {x + 1}^2}\)

Quotient Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {x + 1} \cdot 1 - x \cdot 1} {\paren {x + 1}^2}\)

Power Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\paren {x + 1}^2}\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $6$.