Integral or differential

$\begin{aligned} F(x) & = \int_{x^2}^{4x^2} \sin \sqrt t \ dt & \small \color{#3D99F6} \text{Let } u^2 = t \implies 2u \ du = dt \\ & = 2 \int_x^{2x} u\sin u \ du \\ & = 2 \left(\int_{\color{#3D99F6}a}^{2x} u\sin u \ du - \int_{\color{#3D99F6}a}^x u\sin u \ du \right) & \small \color{#3D99F6} a > 0 \text{ is a constant} \end{aligned}$

$\begin{aligned} \implies F'(x) & = 2 \times \frac d{dx} \left(\int_a^{2x} u\sin u \ du - \int_a^x u\sin u \ du \right) \\ & = 2 \left(2x\sin 2x - x\sin x \right) \end{aligned}$

$\begin{aligned} \implies F' \left(\frac \pi 2\right) & = \boxed{-\pi} \end{aligned}$

$\displaystyle \begin{aligned} F'(x) & = \frac{d}{dx} \int_{x^2}^{4x^2} \sin \sqrt{t} \ dt \\ & = \frac{d}{dx} \left( \int_{x^2}^{c} \sin \sqrt{t} \ dt + \int_{c}^{4x^2} \sin \sqrt{t} \ dt \right) & \small \color{#3D99F6} \text{for some constant } c \\ & = \frac{d}{dx} \left(- \int_{c}^{x^2} \sin \sqrt{t} \ dt + \int_{c}^{4x^2} \sin \sqrt{t} \ dt \right) & \small \color{#3D99F6} \text{Apply the Fundamental Theorem of Calculus} \\ & = -2x \cdot \sin \sqrt{x^2} \ + \ 8x \cdot \sin \sqrt{4x^2} \\ & = -2x \sin |x| \ + \ 8x \sin |2x| \\ & = 2x\left(4\sin |2x| - \sin |x|\right) \\ \implies F'\left(\frac {\pi}{2}\right) & = \boxed{- \pi} \end{aligned}$

We have

$F\left( x \right) = \int _{ x^{ 2 } }^{ 4{ x }^{ 2 } }{ \sin { \sqrt { t } } dt }$ .

Applying Leibniz Rule , we get

$F'\left( x \right) = \sin { \left( 2x \right) \times 8x } - \sin { \left( x \right) \times 2x }$ .

Setting $x = \frac { \pi }{ 2 }$ above , gives

$F'\left( \frac { \pi }{ 2 } \right) = \sin { \left( \pi \right) \times 4\pi } - \sin { \left( \frac { \pi }{ 2 } \right) \times \pi } = \boxed{-\pi}$