# Derivative of Composite Function/Examples/sin(x^2)

## Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {\map \sin {x^2} } = 2 x \map \cos {x^2}$

## Proof

Let $y = x^2$.

Let $z = \sin y$.

Then we have:

$z = \map \sin {x^2}$

and so:

 $\ds \dfrac {\d z} {\d x}$ $=$ $\ds \dfrac {\d z} {\d y} \dfrac {\d y} {\d x}$ Derivative of Composite Function $\ds$ $=$ $\ds \cos y \cdot 2 x$ Derivative of Sine Function, Derivative of Square Function $\ds$ $=$ $\ds 2 x \map \cos {x^2}$ simplification

$\blacksquare$