Image derivative in the Fourier space
The Fourier transform property in your first equation is for continuous time but the image is a discrete time signal. $\partial_x$ isn't exactly differentiation, it's a discrete time filter which does finite differencing.

Since you are only taking derivatives in one direction, let's think about a 1-D signal for simplicity (or, equivalently, let's assume your image has only one row). Let the signal be $f(x)$ and its continuous time Fourier transform $F(u)$. Then we have the transform pair $\frac{df(x)}{dx} \leftrightarrow j 2\pi u F(u)$. In discrete time, if $x[n] \leftrightarrow X[k]$, you can't really say $\{x[n]-x[n-1]\} \leftrightarrow j2\pi k X[k]$.

Instead, we can think of $\partial_x$ as a high pass filter. It then agrees with the intuitive meaning of the continuous time Fourier transform property too (where multiplying by $u$ attenuates lower frequencies more than higher frequencies). Try this code, to see the high pass response:

d = np.zeros(10000)
d[0] = -1.0
d[1] = 1.0
D = np.fft.fft(d)
plt.plot( np.abs(D) )