Derivative of integral defining delta function: Lost minus sign?

Question

I am going through my lecture notes of a theoretical physics class and stumbled across something that irritates me. The line in question is

$$-\frac{\mathrm{d}}{\mathrm{d}x}\int\mathrm{e}^{\mathrm{i}\left(x-y\right)t}\ \mathrm{d}t=2\pi\frac{\mathrm{d}}{\mathrm{d}y}\delta\left(y-x\right)$$

Notice how

1. The minus sign seems to disappear
2. The variable of the derivative changes

I cannot really make sense of this. Is this simply wrong or am I missing something?

Answer 1

From $$ \int e^{i(x-y)t} dt = 2\pi \, \delta(x-y) = 2\pi \, \delta(y-x) $$ we get $$ \frac{\partial}{\partial x} \int e^{i(x-y)t} dt = 2\pi \, \frac{\partial}{\partial x} \delta(y-x) = -2\pi \, \delta'(y-x) = -2\pi \, \frac{\partial}{\partial y} \delta(y-x) . $$