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

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

Solve for f (complex solution)

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

Steps for Solving Linear Equation

The equation is in standard form.

$$yf=\tan(x)$$

Divide both sides by y.

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

Dividing by y undoes the multiplication by y.

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

Solve for f

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

Steps for Solving Linear Equation

The equation is in standard form.

$$yf=\tan(x)$$

Divide both sides by y.

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

Dividing by y undoes the multiplication by y.

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

Solve for x

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

$$x=2\pi n_{2}+\arcsin(\frac{fy}{\sqrt{\left(fy\right)^{2}+1}})\text{, }n_{2}\in \mathrm{Z}\text{, }\exists n_{3}\in \mathrm{Z}\text{ : }\left(n_{3}>\frac{4n_{2}+\frac{2\arcsin(\frac{fy}{\sqrt{\left(fy\right)^{2}+1}})}{\pi }-3}{2}\text{ and }n_{3}<\frac{4n_{2}+\frac{2\arcsin(\frac{fy}{\sqrt{\left(fy\right)^{2}+1}})}{\pi }-1}{2}\right)\text{, }y\neq 0$$