Electronic – Differentiator circuit – can output be larger than input

Did you ever pick up a telescope, hold the wrong end to your eye, and find you can't make sense of what you see?

The only thing wrong with using this telescope to see how an infinite capacitance behaves

\$I=C\frac{dU}{dt}\$

is that you aren't holding the useful end to your eye.

Try it this way round

\$\frac{dU}{dt}=\frac{I}{C}\$

What this equation means is that you can shove any finite amount of current, however large, into an infinite capacitor, and its voltage will not change. This is a convenient approximation for the behaviour of 'very large' capacitors, whose voltage 'does not change much' during operation. LTSpice has an infinite capacitor, and I would guess most other spices do as well. These are useful for approximating coupling or decoupling capacitors, when you are not interested in their deviation from ideal.

Now if you rephrase your question as 'I have a component whose terminal voltage cannot change, and I change its terminal voltage ...', well, you get what you get.

You might want to edit your title so that says 'DC short circuit'.

Hitting the rails is no problem whatsoever. Opamps typically have what's called a push-pull output stage, or some variation of it, which is basically just 2 BJTs configured like so:

When your opamp saturates, that means that one of these BJTs is in cut-off mode (open-circuit) and the other in saturation mode, where it (ideally) acts like a short-circuit, but in practise the voltage between the collector and emitter (Vce) will be at the minimum possible. This will drastically reduce the power consumption of the BJT, which is given by Vce * Current, assuming that the current is the same.

So yes, everything will be fine with your opamp.

But this doesn't mean that it's still desirable to saturate your opamp given your application. Saturation means that your opamp will output a steady voltage while the noise is big enough to keep your opamp in this state. This will introduce a low frequency component into your signal that could fall into the range of frequencies that matter for ECG readings, which is from 1Hz to 12Hz according to a quick google search:

Frequency components from 1 to 12 Hz ( ) are selected for multiple ECG beat recognition. These spectra are not disturbed by high-frequency components above 20 Hz such as power-line interference (50 Hz/60 Hz) and muscle noise, and very low-frequency components (<1 Hz) such as baseline drift and breath.

This might be a problem for your application. If you can, you should try to compute the FFT of your signal to see if the low-frequency harmonics fall into the range of frequencies that you want to read from, and if they do, then add a band-pass filter tuned to the frequencies that you want to read. This will greatly reduce the noise, thus reducing saturation in the opamp, which will also reduce those low frequency harmonics that (may) interfere with your sensor readings.

Take a look at this page to see how to properly implement a band pass filter together with your opamp amplification circuit.