I am taking a theoretical physics module and we are currently going over 4-vectors and tensors, I am finding it hard to get my head around the notion of the covariant derivative. I have been told that: $$\partial=\left(\frac{\partial}{\partial x^0},\frac{\partial}{\partial x^1},\frac{\partial}{\partial x^2},\frac{\partial}{\partial x^3}\right).$$ and hence for any function on spacetime $f(x)$, $$\partial f=\left(\frac{\partial f}{\partial x^0},\frac{\partial f}{\partial x^1},\frac{\partial f}{\partial x^2},\frac{\partial f}{\partial x^3}\right).$$ I have then been given: $$\frac{\partial f}{\partial x^{\mu}}=\sum_{\nu=0}^{3}\frac{\partial f}{\partial x^{'\nu}}\frac{\partial x^{'\nu}}{\partial x^{\mu}}.$$ I am completely lost as to how to get from $\partial f$ to $\frac{\partial f}{\partial x^{\mu}}$.
I was under the impression that $f$, being a function of space time, would be a 4-vector: $$f=\begin{pmatrix}x^0\\x^1\\ x^2 \\x^3 \end{pmatrix},$$ and that: $$\partial f=\begin{pmatrix}\frac{\partial f^0}{\partial x^0} & \frac{\partial f^0}{\partial x^1} & \frac{\partial f^0}{\partial x^2} & \frac{\partial f^0}{\partial x^3}\\\frac{\partial f^1}{\partial x^0} & \frac{\partial f^1}{\partial x^1} & \frac{\partial f^1}{\partial x^2} & \frac{\partial f^1}{\partial x^3}\\ \frac{\partial f^2}{\partial x^0} & \frac{\partial f^2}{\partial x^1} & \frac{\partial f^2}{\partial x^2} & \frac{\partial f^2}{\partial x^3} \\\frac{\partial f^3}{\partial x^0} & \frac{\partial f^3}{\partial x^1} & \frac{\partial f^3}{\partial x^2} & \frac{\partial f^3}{\partial x^3} \end{pmatrix}.$$ This is clearly not the same, I think I my matrix manipulation is the problem, if someone could let me know why it is in the form of a summation, that'd be great. Thanks.