Proof that $\dfrac{\partial U_t}{\partial t} U_t^\dagger$ is anti-Hermitian by integration by parts

Question

A proof to understand why $$\frac{\partial U_t}{\partial t} U_t^\dagger$$ is anti-Hermitian.

Answer 1

Not integration by parts, but the Leibniz rule for products:

Assuming $U_t$ is unitary,

$U_t U_t^\dagger = I; \tag 1$

then

$\dfrac{d}{dt}(U_tU_t^\dagger) = 0, \tag 2$

or

$\dot U_t U_t^\dagger + U_t \dot U_t^\dagger = 0, \tag 3$

so we have

$\dot U_t U_t^\dagger = -U_t \dot U_t^\dagger ; \tag 4$

however,

$(\dot U_t U_t^\dagger)^\dagger = (U_t^\dagger)^\dagger \dot U_t^\dagger = U_t \dot U_t^\dagger, \tag 5$

whence combining (4) and (5) yields

$(\dot U_t U_t^\dagger)^\dagger = -\dot U_t U_t^\dagger, \tag 6$

affirming that $\dot U_t U_t^\dagger$ is anti-Hermitian.