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

Example of Derivative of Composite Function

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

Proof

Let $u = x^2 + x + 1$.

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

Then we have:

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

and so:

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

\(=\)

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

Derivative of Composite Function

\(\ds \)

\(=\)

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

Power Rule for Derivatives

\(\ds \)

\(=\)

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

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $5$.