Derivative of Composite Function/Examples/Root of x + 1 over x - 1

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\sqrt {\dfrac {x + 1} {x - 1} } } = -\dfrac 1 {\sqrt {\paren {x + 1} \paren {x - 1}^3} }$

Proof

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

Let $y = u^{1/2}$.

Thus we have:

$y = \paren {\dfrac {x + 1} {x - 1} }^{1/2}$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

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

Power Rule for Derivatives, Derivative of $\dfrac {x + 1} {x - 1}$

\(\ds \)

\(=\)

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

simplification

\(\ds \)

\(=\)

\(\ds -\dfrac 1 {\sqrt {\paren {x + 1} \paren {x - 1}^3} }\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $18$.