Numerical differentation - discrete data

Question

I was analyzing my data in Origin and I did the numerical differentation. I wanted to find the maximum of $frac {dM}{dT}$. The black curve corresponds to $frac {dM}{dT}$.
My data is discrete and I really don't like the $frac {dM}{dT}$ curve (too spiky).
I got the maximum of $frac {dM}{dT}$ as $T_c = 420K$ which is too much I think. It should be around $390-400$K.
I have a second curve (of the cooling process) which should be the same for the region where I am searching for the maximum and I got a different result ($T_c = 408K$).
I was told that if my curve isn't smooth enough I can get wrong results. How is that? Could it be this case? The differentation curve looks nice for my measurements where I didn't use the heater. It is from a physics experiment but I am interested in maths behind it.

Ross Millikan · Answer

Differentiation is a high pass filter.  If you have a sine wave, like $sin(omega t)$ the derivative is $omega cos (omega t)$.  If you have a number of superposed sine waves it will amplify the high frequencies compared to the low frequencies because of the factor $omega$.  If you just have one trace you can think of Fourier analyzing it to get a superposition of frequencies.  Noise in sampling is the highest frequency you can have because it changes every sample.  All those spikes in the derivative probably came from noise in your data.
Looking at your blue curve, nothing much is happening over an interval less than about $5$ degrees.  You could FFT your data and throw away all the frequencies higher than (say) $1$ per degree, then take the derivative.  A lower tech way to do something similar would be to average all the points over an interval of $1$ or $2$ degrees, then differentiate.  That will smooth some of the noise.

Numerical differentation - discrete data

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