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A differential amplifier and a differentiator are two completely different circuit blocks.
Differential Amplifier
A differential amplifier has two inputs and one output. It takes the difference between the two inputs, multiplies that by the gain, and makes it the output.
Out = (V1 - V2) ⋅ Gain
In this example, the gain is A/B.
Differentiator
A differentiator takes the derivative of a signal. In other words, its output is proportional to how fast the input is changing.
Note that the gain is not dimensionless, as it is for a normal amplifier. For example, the gain can be the output Volts divided by the input Volts/second, which comes out to units of seconds.
In this example, the gain is proportional to -R1⋅C1.
Why do you think the gain could not expressed as V/V?
I'll assume this is referring to the differentiator, since the gain of the differential amplifier is a voltage divided by a voltage, resulting in a dimensionless value.
For a differentiator, the output is the change in the input. Just dividing the output voltage by the input voltage doesn't yield anything meaningful. For example, you get 0 V out for any steady input voltage. Saying you get 0 V out for 10 V in, but also 0 V out for 3.97 V in (or any other voltage), isn't very useful.
What about Acl=-Aol/(1+Aol*beta) with beta=1/(1+sR1C1)?
Since you didn't define any of your terms, nor the context, it's just meaningless characters.
I need to know what is the difference between differential amplifier and differentiator
A differential amplifier amplifies the difference voltage between two signal voltages. A differentiator performs a type of mathematical calculus on a signal. The two processes are wholly unrelated.
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Both differential amplifier and differentiator react to a voltage difference. But in the differential amplifier, the difference is between two voltages applied to the amp inputs at the same time while in the differentiator, the difference is between two voltage values at adjacent moments of time.
I have met a similar question about the difference between a differential amplifier and differential resistance. And in this case, what they have in common, is the voltage difference. But while in the differential amplifier the difference is between two input voltages, in the differential resistance, the difference is between two voltage values at adjacent values of the current.
BTW there is a differential integrator - a 2-input op-amp circuit with two RC circuits. Maybe, it is possible to construct in a similar way a 2-input differential differentiator...
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