So I have a circuit like this:
With the output equation as: \$ V_{out} = -(\frac{R_2}{R_1})\frac{s}{s+\frac{1}{R_1C}} V_{in} \$
So the question is to derive the time-domain equation in terms of the input and also to show that the circuit performs the function of a differentiator.
So here's my work:
\[ V_{out} = -(\frac{R_2}{R_1})\frac{s}{s+\frac{1}{R_1C}} V_{in} \]
\[ \mathcal{L^{-1}}(V_{out}) = -(R_2C)\mathcal{L^{-1}}(\frac{s}{sR_1C+1} V_{in}) \]
If we assume that \$R_1=0\$ then we have
\[ \mathcal{L^{-1}}(V_{out}) = -(R_2C)\mathcal{L^{-1}}(s V_{in}) \]
The \$sV_{in}\$ implies a differentiator since \$ \mathcal{L^{-1}}(s) = \delta'(t) \$
But I'm pretty sure that is either totally wrong or mostly wrong. Where am I going wrong on this derivation for the time-domain equation?