# Derivative of Composite Function/Examples/Root of 1 + x

## 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

$\blacksquare$