ແກ້ fy=tanx.u=v-1/v.text{and}v=lnx.text{whatisthevalueof}dy/dxtext{at}x=e?

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

ແກ້ສຳລັບ f (complex solution)

$$\left\{\begin{matrix}f=\frac{\tan(x)}{y}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\text{ and }y\neq 0\\f\in \mathrm{C}\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }x=\pi n_{2}\text{ and }y=0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\end{matrix}\right,$$

ຂັ້ນຕອນສຳລັບການແກ້ສົມຜົນເສັ້ນຊື່

ສົມຜົນຢູ່ໃນຮູບແບບມາດຕະຖານ.

$$yf=\tan(x)$$

ຫານທັງສອງຂ້າງດ້ວຍ y.

$$\frac{yf}{y}=\frac{\tan(x)}{y}$$

ການຫານດ້ວຍ y ຈະຍົກເລີກການຄູນດ້ວຍ y.

$$f=\frac{\tan(x)}{y}$$

ແກ້ສຳລັບ f

$$\left\{\begin{matrix}f=\frac{\tan(x)}{y}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\text{ and }y\neq 0\\f\in \mathrm{R}\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }x=\pi n_{2}\text{ and }y=0\end{matrix}\right,$$

ຂັ້ນຕອນສຳລັບການແກ້ສົມຜົນເສັ້ນຊື່

ສົມຜົນຢູ່ໃນຮູບແບບມາດຕະຖານ.

$$yf=\tan(x)$$

ຫານທັງສອງຂ້າງດ້ວຍ y.

$$\frac{yf}{y}=\frac{\tan(x)}{y}$$

ການຫານດ້ວຍ y ຈະຍົກເລີກການຄູນດ້ວຍ y.

$$f=\frac{\tan(x)}{y}$$

ແກ້ສຳລັບ x

$$x=\pi +2n_{3}\pi +arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})\text{, }n_{3}\in \mathrm{Z}\text{, }\exists n_{46}\in \mathrm{Z}\text{ : }\left(n_{3}>\left(-\frac{1}{2}\right)\left(\frac{1}{2}\pi +arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})+\left(-1\right)\pi n_{46}\right)\pi ^{-1}\text{ and }n_{3}<\left(-\frac{1}{2}\right)\left(\left(-\frac{1}{2}\right)\pi +arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})+\left(-1\right)\pi n_{46}\right)\pi ^{-1}\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\pi +2n_{3}\pi +arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})=\frac{1}{2}\pi +\pi n_{1}$$

$$x=arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})+2\pi n_{24}\text{, }n_{24}\in \mathrm{Z}\text{, }\exists n_{46}\in \mathrm{Z}\text{ : }\left(n_{46}<\left(arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})+2\pi n_{24}+\left(-\frac{1}{2}\right)\pi \right)\pi ^{-1}\text{ and }n_{46}>\pi ^{-1}\left(arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})+2\pi n_{24}+\left(-\frac{3}{2}\right)\pi \right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }arcSin(fy\left(f^{2}y^{2}+1\right)^{-\frac{1}{2}})+2\pi n_{24}=\frac{1}{2}\pi +\pi n_{1}$$