Derivative of Composite Function/Examples/Root of x^2 + x + 1
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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 $\text {IX}$: $5$.