I am required to show that the operator $\partial_t$ is Anti-Hermitian. This operator is defined such that
$$\partial_t: s(t) \rightarrow \partial_t s(t) $$
Where the definition of an Anti-Hermitian operator in terms of the inner product is
$$<s_1, As_2> = - <As_1, s_2>$$
using this definition of the inner product:
$$<s_1, s_2> = \int_{-\infty}^{+\infty} s_1(t)[s_2(t)]^{*}dt$$
where $^{*}$ denotes the conjugate. This is probably quite a simple question (I am told) but I can't quite seem to make it make much sense. Is it a matter of using integration by parts? I tried but couldn't get the negative sign to appear from anywhere. Maybe I'm slightly confused about the purpose of the complex conjugate in these Fourier transforms.
Any help much appreciated
It is simply using integration by parts. Note that boundary terms must vanish otherwise the integral wouldn't converge. Then note that the derivative commutes with conjugation.