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

Example of Derivative of Composite Function

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

Let $u = \sin x$.

Let $y = u^2$.

Then we have:

$y = \paren {\sin x}^2$

and so:

$\ds \dfrac {\d y} {\d x}$

$=$

$\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}$

$\ds $

$=$

$\ds 2 u \cdot \cos x$

$\ds $

$=$

$\ds 2 \sin x \cos x$

substituting for $u$

$\blacksquare$

2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): chain rule
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): chain rule