In his famous paper on Special Relativity, Einstein derives the Lorentz Transformations. He considers a light beam emitted at time $t$ from the origin of the system of coordinates $k$ towards a point that moves with the origin of the system $K$ such that its coordinate on the $K$ system is $x'=x-vt$ and is then reflected back. He begins with the equation $$\frac{1}{2}[\tau(0,0,0,t)+\tau(0,0,0,t+\frac{x'}{c-v}+\frac{x'}{c+v})]=\tau(x',0,0,\frac{x'}{c-v})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(1)$$ Then the paper says "Hence, if $x'$ be chosen infinitesimally small" $$\frac{1}{2}(\frac{1}{c-v}+\frac{1}{c+v})\frac{\partial\tau}{\partial t}=\frac{\partial\tau}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(2)$$ which is simplified to $$\frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(3)$$ I have read and know how to go from equation (1) to equation (2) using differentials and partial derivatives, but recently, I found a forum thread which stated that what Einstein means by "Making $x'$ infinitely small" is to take a Taylor Series of the components of equation (1) and reducing $x'$ to $0$. Yet, I am not sure of how to do that. $$$$ Can someone help me?

In his famous paper on Special Relativity, Einstein derives the Lorentz Transformations. He considers a light beam emitted at time $t$ from the origin of the system of coordinates $k$ towards a point that moves with the origin of the system $K$ such that its coordinate on the $K$ system is $x'=x-vt$ and is then reflected back. He begins with the equation

$$ \frac{1}{2}\left[\tau(0,0,0,t)+\tau\left(0,0,0,t+\frac{x'}{c-v}+\frac{x'}{c+v}\right)\right]=\tau\left(x',0,0,\frac{x'}{c-v}\right)\tag{1} $$

Then the paper says "Hence, if $x'$ be chosen infinitesimally small"

$$ \frac{1}{2} \left(\frac{1}{c-v}+\frac{1}{c+v}\right)\frac{\partial\tau}{\partial t}=\frac{\partial\tau}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}\tag{2} $$

which is simplified to

$$ \frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}=0\tag{3} $$ I have read and know how to go from equation (1) to equation (2) using differentials and partial derivatives, but recently, I found a forum thread which stated that what Einstein means by "Making $x'$ infinitely small" is to take a Taylor Series of the components of equation (1) and reducing $x'$ to $0$. Yet, I am not sure of how to do that. Can someone help me?

In his famous paper on Special Relativity, Einstein derives the Lorentz Transformations. He considers a light beam emitted at time $t$ from the origin of the system of coordinates $k$ towards a point that moves with the origin of the system $K$ such that its coordinate on the $K$ system is $x'=x-vt$ and is then reflected back. He begins with the equation $$\frac{1}{2}[\tau(0,0,0,t)+\tau(0,0,0,t+\frac{x'}{c-v}+\frac{x'}{c+v})]=\tau(x',0,0,\frac{x'}{c-v})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(1)$$ Then the paper says "Hence, if $x'$ be chosen infinitesimally small" $$\frac{1}{2}(\frac{1}{c-v}+\frac{1}{c+v})\frac{\partial\tau}{\partial t}=\frac{\partial\tau}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(2)$$ which is simplified to $$\frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(3)$$ I have read and know how to go from equation (1) to equation (2) using differentials and partial derivatives, but recently, I found a forum thread which stated that what Einstein means by "Making $x'$ infinitely small" is to take a Taylor Series of the components of equation (1) and reducing $x'$ to $0$. Yet, I am not sure of how to do that. $$$$ Can someone help me?

In his famous paper on Special Relativity, Einstein derives the Lorentz Transformations. He considers a light beam emitted at time $t$ from the origin of the system of coordinates $k$ towards a point that moves with the origin of the system $K$ such that its coordinate on the $K$ system is $x'=x-vt$ and is then reflected back. He begins with the equation $$\frac{1}{2}[\tau(0,0,0,t)+\tau(0,0,0,t+\frac{x'}{c-v}+\frac{x'}{c+v})]=\tau(x',0,0,\frac{x'}{c-v})\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(1)$$ Then the paper says "Hence, if $x'$ be chosen infinitesimally small" $$\frac{1}{2}(\frac{1}{c-v}+\frac{1}{c+v})\frac{\partial\tau}{\partial t}=\frac{\partial\tau}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(2)$$ which is simplified to $$\frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(3)$$ I have read and know how to go from equation (1) to equation (2) using differentials and partial derivatives, but recently, I found a forum thread which stated that what Einstein means by "Making $x'$ infinitely small" is to take a Taylor Series of the components of equation (1) and reducing $x'$ to $0$. Yet, I am not sure of how to do that. $$$$ Can someone help me?