Résoudre y=tanx.u=v-1/v.text{and}v=lnx.text{whatisthevalueof}dy/xtext{at}x=e?

$$y = \tan x . u = v - \frac { 1 } { v } . \text { and } v = \ln x . \text { what is the value of } \frac { d y } { x } \text { at } x = e ?$$

Calculer x

$$x=2\pi n_{1}+\arcsin(\frac{y}{\sqrt{y^{2}+1}})+\pi \text{, }n_{1}\in \mathrm{Z}\text{, }\exists n_{3}\in \mathrm{Z}\text{ : }\left(n_{1}>\frac{2n_{3}-\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }-1}{4}\text{ and }n_{1}<\frac{2n_{3}-\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }+1}{4}\right)$$

$$x=2\pi n_{2}+\arcsin(\frac{y}{\sqrt{y^{2}+1}})\text{, }n_{2}\in \mathrm{Z}\text{, }\exists n_{3}\in \mathrm{Z}\text{ : }\left(n_{3}>\frac{4n_{2}+\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }-3}{2}\text{ and }n_{3}<\frac{4n_{2}+\frac{2\arcsin(\frac{y}{\sqrt{y^{2}+1}})}{\pi }-1}{2}\right)$$

Calculer y

$$y=\tan(x)$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}$$