Why taking derivative amplifies noise

Did you know?
Taking derivative amplifies noise.

Why?
By differentiating, you remove the low-frequency components

How?
Assume a noisy input signal

Taking derivative using diff(inputSig)

When?
1) Finding velocity of motor through its quantized signals (Encoder data or A/D converter output)
2) Image processing, where derivatives are used to help detect the edges. While at mean time, this operation would also make the data noisier

Solution?
As 2 point discrete differentiation is bound to produce highly noisy results. try either
0) Smoothing filter (such as linear phase FIR filter) to alleviate the noise, then do the derivatives
1) 5-points stencil (https://en.wikipedia.org/wiki/Five-point_stencil)
2) Generate coefficients (i.e. more points) yourself using derivation of Lagrange polynomials.
3) Wavelet transform and use derivatives of wavelets. Wavelet transform will allow one to discard high-frequency components, theoretically coming from the underlying noise and sampling rate
( https://www.sciencedirect.com/science/article/abs/pii/S0169743903001370 )