I'm trying to find an operator that will extract the quantity $\frac{dx(t)}{dt}$ from the function $\delta(y - x(t))$. My attempt is to get something close and then subtract the part that isn't the expression. For example I tried this operator $$\frac{d}{dt}y$$ Applying this operator to the delta function gives $$\frac{d}{dt}(y \delta(y - x(t)))$$ Using the picking property of the delta function $$\frac{d}{dt}(x(t) \delta(y - x(t)))$$ Using the chain rule $$\frac{dx(t)}{dt} \delta(y - x(t)) + \frac{d\delta(y - x(t))}{dt} x(t)$$ The goal is to get to this $$\hat{D}\delta(y - x(t)) = \frac{dx(t)}{dt} \delta(y - x(t))$$ I already have this expression in one of my terms. I was thinking of subtracting this term off with another operator so I tried this $$y \frac{d}{dt} $$ However applying this on the delta function doesn't give back the second term I want to get rid of.
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Ive been messing around with other operators the most promising one is $$\frac{d}{dt}$$ Applying this on the delta function gives $$\frac{d\delta(y - x(t))}{dt}$$ Using the chain rule this becomes $$\frac{dx}{dt}\frac{d\delta(y - x(t))}{dx}$$ Is there any identities or something I can use to remove the derivative from the delta function. Maybe a modified form of the identity: $$\frac{d\delta(y - x)}{dy} f(y) = f(x)\delta ' (y - x) - f'(x)\delta(y - x)$$ Many Thanks Josh.