For the differentiator, an input capacitor was used so as to block
constant signals & just output the rate of change. Some examples of
calculated derivatives where for constantly changing which resulted in a
constant, & the sinusoidal wave which resulted in a cosinusoidal
output, which is just a phase shifted sine wave.
To understand the differentiator's use as a high pass filter, we are
going to focus on this last derivative & combine with our
understanding of capacitive reactance.
Starting with DC & very low frequencies, the reactance of the
capacitor becomes essentially infinite, since it blocks all current due
to the voltage buildup inside of it. This makes the gain equation of the
inverting amplifier it is based on to approach zero.
Vout = Vin (-Rf / Rin)
As the frequency increases, less residual charge stays in the capacitor
making it less restrictive to the apparent current flow, which results
in less reactance, driving the ratio of resistances higher as the
reactance approaches zero.
At very high frequencies, the capacitive reactance becomes so low that
it is essentially a closed switch, drawing large amounts of current that
need to be compensated by the opamp, which reaches saturation on each
semicycle of the input signal; At high frequencies the gain approaches
infinity.
To limit the gain at high frequencies, a resistor is used in series with
the input capacitor. What this does is that as the capacitive reactance
gets lower to the point of approaching zero, the series resistance
becomes the dominant component that prevents the flow of current,
limiting the gain to the ratio of that input resistor & the output
resistor, just like a simple inverting amplifier.
So you see, the differentiator also works as a high pass filter, being
the inverse operation in both mathematical terms as the integrator (a
derivative is the inverse operation of the integral) & in filter
functionality (blocks the opposite side of the frequencies)
