Showing the Minkowski force in the covariant form

So I would like to show

begin{equation}
K^{mu} = frac{q}{c}F^{munu}u_{nu} = frac{q}{c}left[partial^{mu}(A^{nu}u_{nu}) – frac{dA^{mu}}{d tau}right]
end{equation}

and I went about as follows:

I can re-write the electric and magnetic fields of

begin{equation}
textbf{E}(textbf{r},t) = -nablaPhi(textbf{r},t) -frac{1}{c}frac{partial textbf{A}(textbf{r},t)}{partial t}
end{equation}

begin{equation}
textbf{B}(textbf{r},t) = nabla times textbf{A}(textbf{r},t)
end{equation}

by employing the four-gradient notation to get

begin{equation}
E^i = -(partial^0A^i – partial^iA^0)
end{equation}

begin{equation}
B^i = -(partial^jA^k – partial^kA^j)
end{equation}

where the indices $(ijk)$ are cyclic permutations of $(123)$. I can find the contravariant electromagnetic field tensor by putting these expressions for the electric and magnetic fields together in matrix form as

begin{equation}
F^{munu} = partial^{mu}A^{nu} – partial^{nu}A^{mu} =
begin{bmatrix}
0 & -E_x & -E_y & -E_z
E_x & 0 & -B_z & B_y
E_y & B_z & 0 & -B_x
E_z & -B_y & B_x & 0
end{bmatrix}
end{equation}

If I multiply this tensor with the four-velocity I get

begin{equation*}
begin{aligned}
F^{munu}u_{nu} = gamma
begin{bmatrix}
0 & -E_x & -E_y & -E_z
E_x & 0 & -B_z & B_y
E_y & B_z & 0 & -B_x
E_z & -B_y & B_x & 0
end{bmatrix}
begin{bmatrix}
c
-dot{x}
-dot{y}
-dot{z}
end{bmatrix}
=
gamma
begin{bmatrix}
textbf{E} cdot textbf{r}
cE_x + dot{y}B_z – dot{z}B_y
cE_y + dot{z}B_x – dot{x}B_z
cE_z + dot{x}B_y – dot{y}B_x
end{bmatrix}
end{aligned}
end{equation*}

which gives me

begin{equation}
F^{munu}u_{nu} =
begin{bmatrix}
textbf{E} cdot dot{textbf{r}}
ctextbf{E} + dot{textbf{r}} times textbf{B}
end{bmatrix}
end{equation}

I notice that the space component of the right-hand side is similar to the Lorentz force (minus some factors)

begin{equation}
textbf{F}_L = etextbf{E} + frac{e}{c}dot{textbf{r}} times textbf{B}
end{equation}

I can find from this the form-invariant generalization of Newton’s second law for this case as

begin{equation}
frac{e}{c}F^{munu}u_{nu} = frac{d(mu^{mu})}{dtau}
end{equation}

Now my question is this: I’ve been told the reasoning in the next step where I say that before I write the Minkowski force given by the contracted field tensor I note that:

begin{equation*}
begin{aligned}
(partial^{mu}A^{nu})u_{nu} = partial^{mu}(A^{nu}u_{nu}) – A^{nu}underbrace{partial^{nu}u_{nu}}_text{=0}
end{aligned}
end{equation*}

and

begin{equation*}
begin{aligned}
frac{dA^{mu}}{dtau} = (partial^{nu}A^{mu})frac{dx_{nu}}{dtau} + underbrace{frac{partial A^{mu}}{partial tau}}_text{=0}
end{aligned}
end{equation*}

My reasoning for this is that the respective partial derivatives vanish because $u_{nu}$ is not an explicit function of the spacial component $x_{mu}$ and $A^{mu}$ is not an explicit function of the time component $tau$. I have been told however that there is an error in my argument here. From this reasoning I went on to say that I can write the Minkowski force as

begin{equation*}
begin{aligned}
K^{mu} = frac{e}{c}F^{munu}u_{nu} = frac{e}{c}left[partial^{mu}(A^{nu}u_{nu}) – frac{dA^{mu}}{dtau}right]
end{aligned}
end{equation*}

but that argument would not hold if my reasoning is incorrect.

tl;dr: Apparently my reasoning to get from

begin{equation}
frac{e}{c}F^{munu}u_{nu} = frac{d(mu^{mu})}{dtau}
end{equation}

to

begin{equation*}
begin{aligned}
K^{mu} = frac{e}{c}F^{munu}u_{nu} = frac{e}{c}left[partial^{mu}(A^{nu}u_{nu}) – frac{dA^{mu}}{dtau}right]
end{aligned}
end{equation*}

is wrong, but I’m not seeing the problem. Does anyone on the stackexchange see my problem?