# How do you differentiate f(x) = sec(tan(sec(tan(x))))?

Jun 5, 2015

$f \left(x\right) = \sec \left\{\tan \left[\sec \left(\tan x\right)\right]\right\}$

If you write this as:
$f \left\{g \left[h \left(i \left(x\right)\right)\right]\right\}$

then

$f ' \left(x\right) = f ' \left\{g \left[h \left(i \left(x\right)\right)\right]\right\} \cdot g ' \left[h \left(i \left(x\right)\right)\right] \cdot h ' \left[i \left(x\right)\right] \cdot i ' \left(x\right)$

and

$f ' \left(\sec u\right) = \left(\sec u \tan u\right) \cdot u ' \left(x\right)$
and
$f ' \left(\tan u\right) = \left({\sec}^{2} u\right) \cdot u ' \left(x\right)$

so

$f ' \left(x\right) =$
$\sec \left\{\tan \left[\sec \left(\tan x\right)\right]\right\} \tan \left\{\tan \left[\sec \left(\tan x\right)\right]\right\}$
$\cdot {\sec}^{2} \left(\sec \left(\tan x\right)\right)$
$\cdot \sec \left(\tan x\right) \tan \left(\tan x\right)$
$\cdot {\sec}^{2} x$