UndergraduateSTEMReasoning

{
  "id": "6a6ac0c02a9a00a015e116cd",
  "subject": "Mathematics",
  "level": "Undergraduate",
  "qtype": "Calculation",
  "question": "Let $g(x)$ and $h(x)$ be differentiable functions, and let $f(x,t)$ be a differentiable function with respect to its first variable. Define the function $$F(x) = \\int_{g(x)}^{h(x)} f(x,t) \\, dt.$$ Find the derivative $F'(x)$ in terms of $f$, its partial derivative with respect to $x$, $g$, $h$, and their derivatives.",
  "options": null,
  "answer": "$-f(x,g(x))g'(x) + f(x,h(x))h'(x) + \\int_{g(x)}^{h(x)} \\frac{\\partial f}{\\partial x}(x,t) \\, dt$",
  "solution": "To compute $F'(x)$ efficiently, we treat the integral as a composite function of three independent variables. Define\n$$\\Phi(u,v,w) = \\int_u^v f(w,t)\\,dt.$$\nThen the original function can be written as $F(x) = \\Phi(g(x), h(x), x)$. By the multivariable chain rule:\n$$F'(x) = \\frac{\\partial \\Phi}{\\partial u} g'(x) + \\frac{\\partial \\Phi}{\\partial v} h'(x) + \\frac{\\partial \\Phi}{\\partial w} \\cdot 1.$$\nUsing the Fundamental Theorem of Calculus, we compute the partial derivatives of $\\Phi$:\n$$\\frac{\\partial \\Phi}{\\partial u} = -f(w,u), \\quad \\frac{\\partial \\Phi}{\\partial v} = f(w,v), \\quad \\frac{\\partial \\Phi}{\\partial w} = \\int_u^v \\frac{\\partial f}{\\partial x}(w,t)\\,dt.$$\nSubstituting $u=g(x)$, $v=h(x)$, and $w=x$ into these partials and then into the chain rule expression gives:\n$$F'(x) = -f(x, g(x))g'(x) + f(x, h(x))h'(x) + \\int_{g(x)}^{h(x)} \\frac{\\partial f}{\\partial x}(x,t)\\,dt.$$\nThis elegantly derives the Leibniz integral rule without needing to split the integral at a constant base point."
}