F(x) = ∫a，x(x-t)f'(t)dt

= ∫a，xxf'(t)dt - ∫a，xtf'(t)dt

=x ∫a，xf'(t)dt - ∫a，xtf'(t)dt

Bilateral derivation:

F'(x) = (x)' ∫a，xf'(t)dt + x(∫a，xf'(t)dt )' - (∫a，xtf'(t)dt)'

= ∫a，xf'(t)dt + xf'(x) - xf'(x)

= ∫a，xf'(t)dt

It seems that f(t) is also typed as f'(t), otherwise how could it be A'?