Why taking derivative amplifies noise In image processing we may use derivatives to help us detect the edges. While at mean time, this operation would also make the data noisier. But why do we have this effect?

My intuition is that if we denote the image data as

$I_{actual}(x, y) = I_{ideal}(x, y) + I_{noise}(x, y)$.

Then apparently when we do $I_{actual}(x_2, y) - I_{actual}(x_1, y)$, the noises will cancel out with each other? But how come they end up adding up to each other?
Differentiation in one axis will amplify high-frequency components. Assuming noise at Fs/2 (i.e maximum frequency), the gain at this frequency will be 6 dB. Intuitively, if you have a sequence alternating between +1 and -1 i.e (1, -1, 1, - 1), Then the difference will be +2, - 2, +2, -2, a gain of 6 dB

Another reason is that by differentiating, you remove the low-frequency components, so your eyes see mostly the high-frequency components giving you the impression of having more noise.