Derivative of an accumulation function with variable lower limit

First, reverse the limits of integration to write $F(x)$ with the variable at the top: $F(x) = \int_{2x}^{7} \sqrt{t^3+1}\,dt = -\int_{7}^{2x} \sqrt{t^3+1}\,dt.$ This step is necessary because FTC Part 1 is stated for integrals of the form $\int_a^{g(x)} f(t)\,dt$, with the variable limit on top.

Now apply FTC Part 1 with $g(x) = 2x$: $\frac{d}{dx}\int_{7}^{2x}\sqrt{t^3+1}\,dt = \sqrt{(2x)^3+1}\cdot \frac{d}{dx}(2x) = 2\sqrt{8x^3+1}.$ Multiplying by the overall negative sign from the swap gives $F'(x) = -2\sqrt{8x^3+1}.$