For the differentiator, an input capacitor was used so as to block constant signals and just output the rate of change. Some examples of calculated derivatives where for constantly changing which resulted in a constant, and 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 and combine with our understanding of capacitive reactance.
Starting with DC and 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 and 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) and in filter functionality (blocks the opposite side of the frequencies).
