Leibniz Rule

When the upper limit of an accumulation function is itself a function of x, the plain Fundamental Theorem is not enough and the Leibniz rule takes over by layering the chain rule on top. The governing formula is \frac{d}{dx}\int_0^{g(x)} f(t),dt = f(g(x))\cdot g'(x), where the integrand is evaluated at the moving boundary and then scaled by the boundary's rate of change. Here f(t) = \sin t and g(x) = x^2, so g'(x) = 2x. Substituting into the formula, the derivative equals \sin(x^2)\cdot 2x = 2x\sin(x^2). The chain-rule factor 2x is essential precisely because the upper limit varies nonlinearly with x. Option \sin(x^2) forgets the chain-rule multiplier coming from the variable upper limit. Option 2x\cos(x^2) wrongly differentiates the integrand into a cosine rather than merely evaluating it at the limit. Option \cos(x^2) commits both of those errors at once. This rule generalizes accumulation to moving boundaries, a recurring JEE Advanced configuration. As a final plausibility check, the 2x factor must appear because the upper limit grows quadratically, accelerating the accumulation of area, and the multiplier captures that acceleration consistently for all real x.