Derivative of Composite Function/Examples/Root of x^2 + 1

Example of Derivative of Composite Function

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

Proof

Let $u = x^2 + 1$.

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

Thus we have:

$y = \sqrt {x^2 + 1}$

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

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

Power Rule for Derivatives, Sum Rule for Derivatives

\(\ds \)

\(=\)

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

Power Rule for Derivatives, Derivative of Constant

\(\ds \)

\(=\)

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

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $27$.