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