Differentiator Circuit
The most straightforward way to work this with R1 = 0 is to see that the time-domain current through the capacitor is just:

\$i_C = C \dfrac{dv_c}{dt} = C\dfrac{dv_{in}}{dt} \$

due to the virtual ground at pin 2 of the (ideal) op amp.

It follows that:

\$v_{out} = -R_2C\dfrac{dv_{in}}{dt} \$

Now, the circuit pictured is not a differentiator; it is a high-pass filter.  However, when the frequency is much lower than \$\frac{1}{R_1C} \$, the frequency-domain output is approximately:

\$V_{out} = -R_2CsV_{in} \$

But differentiation in the time domain is multiplication by s in the complex frequency domain.  So, the circuit is an approximation of a differentiator when the input frequencies of interest are much lower than \$\frac{1}{R_1C} \$.