What is the derivative of f(x) = sin (cos (tanx) )?

Mar 13, 2018

$f ' \left(x\right) = - {\sec}^{2} x \sin \left(\tan x\right) \cos \left(\cos \left(\tan x\right)\right)$

Explanation:

$f \left(x\right) = \sin \left(g \left(x\right)\right)$
$f ' \left(x\right) = g ' \left(x\right) \cos \left(g \left(x\right)\right)$

$g \left(x\right) = \cos \left(h \left(x\right)\right)$
$g ' \left(x\right) = - h ' \left(x\right) \sin \left(h \left(x\right)\right)$

$h \left(x\right) = \tan \left(x\right)$
$h ' \left(x\right) = {\sec}^{2} x$

$g ' \left(x\right) = - {\sec}^{2} x \sin \left(\tan x\right)$
$g \left(x\right) = \cos \left(\tan x\right)$

$f ' \left(x\right) = - {\sec}^{2} x \sin \left(\tan x\right) \cos \left(\cos \left(\tan x\right)\right)$