# What is the derivative of "arcsec"(e^(2x))?

May 19, 2018

$\frac{d}{\mathrm{dx}} \left[\arccos \left(\frac{1}{e} ^ \left(2 x\right)\right)\right] = - \frac{1}{\sqrt{1 - {\left(\frac{1}{e} ^ \left(2 x\right)\right)}^{2}}} \cdot \left[- \frac{2}{e} ^ \left(2 x\right)\right]$

$= \frac{\frac{2}{e} ^ \left(2 x\right)}{\sqrt{1 - {\left(\frac{1}{e} ^ \left(2 x\right)\right)}^{2}}} = \frac{2}{{e}^{2 x} \cdot \sqrt{1 - {\left(\frac{1}{e} ^ \left(2 x\right)\right)}^{2}}}$

#### Explanation:

there two different method to derive $a r c \sec \left({e}^{2 x}\right)$

i will use one of them

$a r c \sec \left({e}^{2 x}\right) = \arccos \left(\frac{1}{e} ^ \left(2 x\right)\right)$

$\frac{d}{\mathrm{dx}} \left[\arccos u\right] = - \frac{1}{\sqrt{1 - {\left(u\right)}^{2}}} \cdot u '$

$\frac{d}{\mathrm{dx}} \left[\arccos \left(\frac{1}{e} ^ \left(2 x\right)\right)\right] = - \frac{1}{\sqrt{1 - {\left(\frac{1}{e} ^ \left(2 x\right)\right)}^{2}}} \cdot \left[- \frac{2}{e} ^ \left(2 x\right)\right]$

$= \frac{\frac{2}{e} ^ \left(2 x\right)}{\sqrt{1 - {\left(\frac{1}{e} ^ \left(2 x\right)\right)}^{2}}} = \frac{2}{{e}^{2 x} \cdot \sqrt{1 - {\left(\frac{1}{e} ^ \left(2 x\right)\right)}^{2}}}$