Leibniz Rule: d/dx integral(g(x),h(x)) f(t) dt = f(h(x))*h'(x) - f(g(x))*g'(x)

Application 1: Finding Derivatives of Integral Functions Given F(x) = integral(0 to $x^2$) e^(-$t^2$) dt, find F'(x). F'(x) = e^(-$x^4$) * 2x. The key is applying the chain rule to the upper limit.

Application 2: Solving Integral Equations Given integral(0 to x) f(t) dt = g(x), differentiate: f(x) = g'(x). More complex: integral(0 to x) f(t) dt = x*f(x) + h(x) — differentiate both sides carefully.

Application 3: Finding Extrema If F(x) = integral(a to x) f(t) dt, then F'(x) = f(x) = 0 gives critical points. F''(x) = f'(x) determines max/min.

Application 4: Both Limits Variable F(x) = integral(sinx to cosx) f(t) dt. F'(x) = f(cosx)(-sinx) - f(sinx)(cosx).

JEE Pattern: "If f is continuous and integral(0 to x) f(t) dt = $xe^x$ - integral(0 to x) $e^t$ f(t) dt, find f(x)." Differentiate: f(x) = $e^x$ + $xe^x$ - $e^x$*f(x). Solve: f(x)(1+$e^x$) = $e^x$(1+x).