Derivative of Composite Function/Examples/Root of 1 + x
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Example of Derivative of Composite Function
- $\map {\dfrac \d {\d x} } {\sqrt {1 + x} } = \dfrac 1 {2 \sqrt {1 + x} }$
Proof
Let $u = 1 + x$.
Let $y = u^{1/2}$.
Thus by definition of square root we have:
- $y = \paren {1 + x}^{1/2}$
and so:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\) | Derivative of Composite Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 u^{-1/2} \cdot 1\) | Power Rule for Derivatives, Derivative of Identity Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {2 \sqrt {1 + x} }\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $1$.