Let $F(x) = \int_0^{x^2} \cos(t) \, dt$. Using the Leibniz rule for differentiation of an integral with variable limits, the derivative $\frac{dF}{dx}$ evaluated at $x = \sqrt{\frac{\pi}{2}}$ is:

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### 1. Strategy & Theoretical Principles By the Leibniz integral rule (Fundamental Theorem of Calculus with variable limits): $$\frac{d}{dx} \left[ \int_{a(x)}^{b(x)} f(x, t) \, dt \right] = f(x, b(x)) \frac{db}{dx} - f(x, a(x)) \frac{da}{dx} + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x} \, dt$$ ### 2. Analytical Formulation For $F(x) = \int_0^{x^2} \cos(t^2) \, dt$: - Upper limit $b(x) = x^2 \implies \frac{db}{dx} = 2x$ - Lower limit $a(x) = 0 \implies \frac{da}{dx} = 0$ - The integrand has no explicit dependence on $x$ ($\frac{\partial f}{\partial x} = 0$). Therefore: $$\frac{dF}{dx} = \cos\left((x^2)^2\right) \cdot (2x) - 0 = 2x \cos(x^4)$$ ### 3. Step-by-Step Calculation Evaluate at $x = \sqrt{\frac{\pi}{2}}$: $$x^2 = \frac{\pi}{2} \implies x^4 = \left(\frac{\pi}{2}\right)^2$$ Wait, let's examine $x^4$: if $F(x) = \int_0^{x^2} \cos(t) dt = \sin(x^2)$, then $F'(x) = 2x \cos(x^2)$. Let's check the argument: if integrand is $\cos(t)$, then $F'(x) = 2x \cos(x^2)$. At $x = \sqrt{\pi/2}$, $x^2 = \pi/2$, so $\cos(\pi/2) = 0$. If integrand is $\cos(t^2)$, then at $x = \sqrt[4]{\pi/2}$, $x^4 = \pi/2$. Let's specify the integrand as $\cos(t)$: $$F(x) = \int_0^{x^2} \cos(t) \, dt$$ Then: $$\frac{dF}{dx} = 2x \cos(x^2)$$ At $x = \sqrt{\frac{\pi}{2}}$: $$\frac{dF}{dx} = 2 \sqrt{\frac{\pi}{2}} \cos\left(\frac{\pi}{2}\right) = 2 \sqrt{\frac{\pi}{2}} \times 0 = 0.0$$ This formulation is clean, unambiguous, and tests the Leibniz rule chain differentiation precisely. ### 4. Distractor Forensics - Forgetting the chain rule factor $2x$ still yields 0 at this point, but if evaluated at other points it leads to severe errors. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always multiply by the derivative of the upper limit $\frac{d(x^2)}{dx} = 2x$. - **Standard Reference:** Kreyszig, E., *Advanced Engineering Mathematics*, Chapter 9.

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