Differentiating an accumulation function

The Fundamental Theorem of Calculus, Part 1 states that if $f$ is continuous, then $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$. Here the integrand is $f(t) = t^4 - 2t$, which is a polynomial and therefore continuous everywhere, so the theorem applies directly with $a = -2$.

Since the upper limit of integration is simply $x$ (not a function of $x$), no chain rule is needed. We just replace $t$ with $x$ in the integrand to get $F'(x) = x^4 - 2x$.