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it means, in english, that "the derivative of the integral is the integrand"

I'm confused about the d/dx part in particular...

that is, if
\[F(x)=\int_a^xf(t)dt\] then
\[F'(x)=f(x)\]

notice that
\[F(x)=\int_a^xf(t)dt\] is a function of the variable \(x\) and not of \(t\)

for example, if
\[F(x)=\int_0^x\sin(t)dt\] then
\[F'(x)=\sin(x)\]