I have my attempt but it needs to be checked for logical faults.

- $\tau$(0,0,0,t) = $\tau$(0,0,0,0) + [$\frac{\partial\tau}{\partial x'}$] * (0-0) + [$\frac{\partial\tau}{\partial y}$] * (0-0) + [$\frac{\partial\tau}{\partial z}$] * (0-0) + [$\frac{\partial\tau}{\partial t}$] * (t-0) = $\tau$(0,0,0,0) + t$\frac{\partial\tau}{\partial t}$

- $\tau$(0,0,0,t+$\frac{x'}{c-v}$+$\frac{x'}{c+v}$) = $\tau$(0,0,0,0) + [$\frac{\partial\tau}{\partial x'}$] * (0-0) + [$\frac{\partial\tau}{\partial y}$] * (0-0) + [$\frac{\partial\tau}{\partial z}$] * (0-0) + [$\frac{\partial\tau}{\partial t}$] * (t+$\frac{x'}{c-v}$+$\frac{x'}{c+v}$-0) = 
$\tau$(0,0,0,0) + t$\frac{\partial\tau}{\partial t}$ + $\frac{x'}{c-v}$$\frac{\partial\tau}{\partial t}$ + $\frac{x'}{c+v}$$\frac{\partial\tau}{\partial t}$

- $\tau$(x',0,0,t+$\frac{x'}{c-v}$) = $\tau$(0,0,0,0) + [$\frac{\partial\tau}{\partial x'}$] * (x'-0) + [$\frac{\partial\tau}{\partial y}$] * (0-0) + [$\frac{\partial\tau}{\partial z}$] * (0-0) + [$\frac{\partial\tau}{\partial t}$] * (t+$\frac{x'}{c-v}$-0) = $\tau$(0,0,0,0) + * x'$\frac{\partial\tau}{\partial x'}$ + t$\frac{\partial\tau}{\partial t}$ + $\frac{x'}{c-v}$$\frac{\partial\tau}{\partial t}$   

Plug those Taylor approximations back into equation (1):

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

Distribute and combine

$\tau$(0,0,0,0) + t$\frac{\partial\tau}{\partial t}$ + $\frac{x'}{2(c-v)}$$\frac{\partial\tau}{\partial t}$ + $\frac{x'}{2(c+v)}$$\frac{\partial\tau}{\partial t}$ = $\tau$(0,0,0,0) + x'$\frac{\partial\tau}{\partial x'}$ + t$\frac{\partial\tau}{\partial t}$ + $\frac{x'}{c-v}$$\frac{\partial\tau}{\partial t}$

Subtract $\tau$(0,0,0,0) and t$\frac{\partial\tau}{\partial t}$ from both sides


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

Take the derivative with respect to x' of both sides

$\frac{d}{dx'}$[$\frac{x'}{2(c-v)}$$\frac{\partial\tau}{\partial t}$ + $\frac{x'}{2(c+v)}$$\frac{\partial\tau}{\partial t}$] = $\frac{d}{dx'}$[x'$\frac{\partial\tau}{\partial x'}$ + $\frac{x'}{c-v}$$\frac{\partial\tau}{\partial t}$]

Factor out $\frac{\partial\tau}{\partial t}$ and $\frac{1}{2}$ on the left side

$\frac{1}{2}$[$\frac{1}{c-v}$ + $\frac{1}{c+v}$]$\frac{\partial\tau}{\partial t}$ = $\frac{\partial\tau}{\partial x'}$ + $\frac{1}{c-v}$$\frac{\partial\tau}{\partial t}$

I assume you know how to get from (2) to (3)