How is the interchange of differentiation and integration of a product of delta function and another function in this problem possible? please explain

Question

guys this is the question i come across- to find the value of

$$\int_{-∞}^{∞}e^{-2t} \delta'(t).dt$$

where $ \delta(t) $ is the usual delta function or impulse function. The problem is solved as the given integral is equal to $$ -\int_{-∞}^∞ \frac{d}{dt} e^{-2t}. \delta(t).dt $$ $$= 2e^{-2t}|_{t=0} $$ $$=2$$ Can someone please explain how this is solved?

Answer 1

It's not an interchange of differentiation and integration. It's integration by parts.