Derivative of Composite Function/Examples/Logarithm of x over x + 1

Example of Derivative of Composite Function

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

Proof

Let $u = \dfrac x {x + 1}$.

Let $y = \ln u$.

Thus we have:

$y = \ln \dfrac x {x + 1}$

and so:

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\)

Derivative of Composite Function

\(\ds \)

\(=\)

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

Derivative of Natural Logarithm, Derivative of $\dfrac x {x + 1}$

\(\ds \)

\(=\)

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

simplification

\(\ds \)

\(=\)

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

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $19$.