The inverse function to integration is differentiation, in other words
finding the derivative, which the opamp can also perform. The derivative
is defined as the rate at which the function changes.
By using an input capacitor instead of a resistor, we can accomplish the
same thing. If you remember, a capacitor stores charges in its plates,
when one of them starts accumulating charges, the same charges will be
pushed out from the other plate, as if current was flowing through the
capacitor despite the intrinsic insulating layer.
The capacitor's charges start building up and creating a voltage across
itself in opposition to the charging voltage, thus slowing down the
incoming charges, slowing down the charging process in general. When
enough charges have accumulated, the charges inside the capacitor
completely push away the charges coming from the source, no more charges
enter the capacitor, & because of this no more charges are pushed out
on the other side of the capacitor, so no more apparent flow of current
across the capacitor.
When used as input for a signal, if the signal does not change (like a
DC input), the capacitor will have an initial apparent current through
it as the voltage across it builds up due to incoming charges, & since
the input of the amplifier tries to not draw any current, it will
create a voltage at its output so that the current through the feedback
resistor is the same as the apparent current through the capacitor.
Since the capacitor charges very quickly due to the voltage applied to
it & the fact that there's no current limiting component like a
resistor, the apparent current through the capacitor falls very quickly
as the voltage across it in opposition rises as quickly; the falling
current is also causes the opamp to drive the output voltage less, since
there's less current to compensate for.
Applying a DC input to the differentiator thus creates a spike in input
as well as in output as the capacitor's initial charge is developed, &
then goes back to 0v as there's no more apparent current to compensate
for; Similar to the operation of finding a constant's derivative, which
is always 0.
The fact that there's an initial spike can be mathematically modeled as a
period in which there's a function that rises at a very high rate
(which actually happens, the voltage doesn't just jump from 0v to the DC
input voltage, it rises very rapidly towards it), so its rate of change
is very high for a brief period of time; hence the spike.
As the input voltage stabilizes, its rate of change slows down very
rapidly as well, going towards zero when fully stabilized; this is
reflected in the opamp's output by the fact that as the voltage
stabilizes, the output spike goes down very rapidly towards zero &
stays there.
Now instead of applying a constant input, you can replace it with a constantly changing input.
If the input is increasing at a constant rate, there will be a constant
apparent flow of current through the capacitor, since the voltage
buildup across the capacitor is compensated by the increase in input
signal. Since there's a constant apparent flow of current through the
capacitor, the opamp compensated by setting the output voltage at a
level that will make the feedback resistor draw the same amount of
current, so that the opamp input does not draw it.
Since the amount of apparent current is constant, a constant output
voltage is enough to keep the feedback resistor drawing the current, &
the opamp keeps a constant output at the output.
This mode is very similar to using a resistor with constant dc as the input.
The same is true for a constantly decreasing input voltage; the output
will just be of reversed polarity. To compare with the mathematical
definition of the derivative of a linear variable, the derivative will
be a constant.
This can be expanded to other functions, one of the most widely used
being the sine function. Since the mathematical derivative of the sin(x)
function is cos(x), which is a shifted version of sin(x) by 90 degrees,
when you input a sine input at the differentiator amplifier, the output
will be the same function shifted 90 degrees, in essence, a cosine
function.
