The repeated index \nu means sum over all four coordinates, while \mu labels which transformed coordinate we mean:

    \[ \boxed{ dx'^\mu = \frac{\partial x'^\mu}{\partial x^0}\,dx^0 + \frac{\partial x'^\mu}{\partial x^1}\,dx^1 + \frac{\partial x'^\mu}{\partial x^2}\,dx^2 + \frac{\partial x'^\mu}{\partial x^3}\,dx^3 } \]

Writing out all four equations:

    \[ \begin{aligned} dx'^0 &= \frac{\partial x'^0}{\partial x^0}dx^0 +\frac{\partial x'^0}{\partial x^1}dx^1 +\frac{\partial x'^0}{\partial x^2}dx^2 +\frac{\partial x'^0}{\partial x^3}dx^3, \\[6pt] dx'^1 &= \frac{\partial x'^1}{\partial x^0}dx^0 +\frac{\partial x'^1}{\partial x^1}dx^1 +\frac{\partial x'^1}{\partial x^2}dx^2 +\frac{\partial x'^1}{\partial x^3}dx^3, \\[6pt] dx'^2 &= \frac{\partial x'^2}{\partial x^0}dx^0 +\frac{\partial x'^2}{\partial x^1}dx^1 +\frac{\partial x'^2}{\partial x^2}dx^2 +\frac{\partial x'^2}{\partial x^3}dx^3, \\[6pt] dx'^3 &= \frac{\partial x'^3}{\partial x^0}dx^0 +\frac{\partial x'^3}{\partial x^1}dx^1 +\frac{\partial x'^3}{\partial x^2}dx^2 +\frac{\partial x'^3}{\partial x^3}dx^3. \end{aligned} \]

Each equation is simply the total differential of one coordinate function x'^\mu(x^0,x^1,x^2,x^3). All partial derivatives are evaluated at the same spacetime point.