{
  "derivadas-logaritmicas": [
    {
      "id": "dl-001",
      "topic": "derivada-ln-x",
      "question": "¿Cuál es la derivada de $f(x) = \\ln(x)$?",
      "type": "multiple-choice",
      "options": [
        "$\\frac{1}{x}$",
        "$\\frac{1}{x^2}$",
        "$x$",
        "$e^x$"
      ],
      "correct": 0,
      "difficulty": "facil",
      "hints": [
        "Esta es la fórmula fundamental",
        "El logaritmo natural tiene una derivada muy simple",
        "d/dx[ln(x)] = 1/x"
      ],
      "stepByStep": [
        "📚 **Derivada del logaritmo natural**",
        "",
        "**Fórmula fundamental:**",
        "$$\\frac{d}{dx}[\\ln(x)] = \\frac{1}{x}$$",
        "",
        "🧮 **Demostración (usando definición)**",
        "",
        "Sabemos que $\\ln(x)$ y $e^x$ son funciones inversas:",
        "$$y = \\ln(x) \\iff e^y = x$$",
        "",
        "**Derivación implícita:**",
        "$$\\frac{d}{dx}[e^y] = \\frac{d}{dx}[x]$$",
        "",
        "$$e^y \\cdot \\frac{dy}{dx} = 1$$",
        "",
        "$$\\frac{dy}{dx} = \\frac{1}{e^y}$$",
        "",
        "Como $e^y = x$:",
        "$$\\frac{dy}{dx} = \\frac{1}{x}$$",
        "",
        "💡 **Restricción importante**",
        "",
        "**Dominio:** $x > 0$",
        "",
        "El logaritmo natural solo está definido para valores positivos.",
        "",
        "📊 **Interpretación geométrica**",
        "",
        "* En $x = 1$: pendiente = $1/1 = 1$",
        "* En $x = 2$: pendiente = $1/2 = 0.5$",
        "* Cuando $x \\to 0^+$: pendiente $\\to \\infty$",
        "* Cuando $x \\to \\infty$: pendiente $\\to 0$",
        "",
        "✅ **Respuesta**",
        "$\\frac{d}{dx}[\\ln(x)] = \\frac{1}{x}$"
      ],
      "explanation": "La derivada del logaritmo natural es 1/x"
    },
    {
      "id": "dl-002",
      "topic": "derivada-ln-x",
      "question": "Calcula $\\frac{d}{dx}[3\\ln(x)]$",
      "type": "multiple-choice",
      "options": [
        "$\\frac{3}{x}$",
        "$\\frac{1}{3x}$",
        "$3x$",
        "$\\ln(3)$"
      ],
      "correct": 0,
      "difficulty": "facil",
      "hints": [
        "Constante sale fuera",
        "d/dx[c·f(x)] = c·f'(x)",
        "= 3 · (1/x)"
      ],
      "stepByStep": [
        "📝 **Derivar: $3\\ln(x)$**",
        "",
        "🧮 **Regla de constante por función**",
        "",
        "$$\\frac{d}{dx}[c \\cdot f(x)] = c \\cdot \\frac{d}{dx}[f(x)]$$",
        "",
        "📐 **Aplicar**",
        "$$\\frac{d}{dx}[3\\ln(x)] = 3 \\cdot \\frac{d}{dx}[\\ln(x)]$$",
        "",
        "$$= 3 \\cdot \\frac{1}{x}$$",
        "",
        "$$= \\frac{3}{x}$$",
        "",
        "💡 **Patrón general**",
        "$$\\frac{d}{dx}[c\\ln(x)] = \\frac{c}{x}$$",
        "",
        "✅ **Respuesta**",
        "$\\frac{3}{x}$"
      ],
      "explanation": "3·d/dx[ln(x)] = 3·(1/x) = 3/x"
    },
    {
      "id": "dl-003",
      "topic": "derivada-ln-x",
      "question": "Si $f(x) = \\ln(x)$, ¿cuál es $f''(x)$ (segunda derivada)?",
      "type": "multiple-choice",
      "options": [
        "$-\\frac{1}{x^2}$",
        "$\\frac{1}{x^2}$",
        "$-\\frac{1}{x}$",
        "$0$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Primera: f'(x) = 1/x = x^(-1)",
        "Segunda: d/dx[x^(-1)]",
        "= -1·x^(-2) = -1/x²"
      ],
      "stepByStep": [
        "📝 **Derivadas sucesivas de ln(x)**",
        "",
        "**Función:**",
        "$$f(x) = \\ln(x)$$",
        "",
        "📐 **Primera derivada**",
        "$$f'(x) = \\frac{1}{x}$$",
        "",
        "Reescribir:",
        "$$f'(x) = x^{-1}$$",
        "",
        "🧮 **Segunda derivada**",
        "$$f''(x) = \\frac{d}{dx}[x^{-1}]$$",
        "",
        "Aplicar regla de la potencia:",
        "$$= -1 \\cdot x^{-2}$$",
        "",
        "$$= -\\frac{1}{x^2}$$",
        "",
        "📊 **Tercera derivada**",
        "$$f'''(x) = \\frac{d}{dx}\\left[-x^{-2}\\right] = 2x^{-3} = \\frac{2}{x^3}$$",
        "",
        "💡 **Patrón general**",
        "$$f^{(n)}(x) = (-1)^{n+1} \\frac{(n-1)!}{x^n}$$",
        "",
        "Para $n \\geq 1$.",
        "",
        "✅ **Respuesta**",
        "$f''(x) = -\\frac{1}{x^2}$"
      ],
      "explanation": "f'(x) = 1/x, f''(x) = -1/x²"
    },
    {
      "id": "dl-004",
      "topic": "derivada-ln-x",
      "question": "¿Cuál es la derivada de $g(x) = x - \\ln(x)$?",
      "type": "multiple-choice",
      "options": [
        "$1 - \\frac{1}{x}$",
        "$\\frac{1}{x} - 1$",
        "$1 + \\frac{1}{x}$",
        "$\\frac{x-1}{x}$"
      ],
      "correct": 0,
      "difficulty": "facil",
      "hints": [
        "Derivar término a término",
        "d/dx[x] = 1",
        "d/dx[ln(x)] = 1/x"
      ],
      "stepByStep": [
        "📝 **Derivar: $x - \\ln(x)$**",
        "",
        "🧮 **Regla de la suma/diferencia**",
        "",
        "$$\\frac{d}{dx}[f(x) \\pm g(x)] = f'(x) \\pm g'(x)$$",
        "",
        "📐 **Aplicar**",
        "$$g'(x) = \\frac{d}{dx}[x] - \\frac{d}{dx}[\\ln(x)]$$",
        "",
        "$$= 1 - \\frac{1}{x}$$",
        "",
        "💡 **Forma alternativa**",
        "$$g'(x) = \\frac{x - 1}{x}$$",
        "",
        "📊 **Punto crítico**",
        "",
        "**g'(x) = 0 cuando:**",
        "$$1 - \\frac{1}{x} = 0$$",
        "",
        "$$\\frac{1}{x} = 1$$",
        "",
        "$$x = 1$$",
        "",
        "Punto crítico en $x = 1$.",
        "",
        "✅ **Respuesta**",
        "$1 - \\frac{1}{x}$"
      ],
      "explanation": "d/dx[x - ln(x)] = 1 - 1/x"
    },
    {
      "id": "dl-005",
      "topic": "derivada-log-base-a",
      "question": "¿Cuál es la derivada de $f(x) = \\log_a(x)$ (donde $a > 0$, $a \\neq 1$)?",
      "type": "multiple-choice",
      "options": [
        "$\\frac{1}{x \\ln(a)}$",
        "$\\frac{\\ln(a)}{x}$",
        "$\\frac{1}{x}$",
        "$\\frac{1}{x \\log(a)}$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Usa cambio de base",
        "log_a(x) = ln(x)/ln(a)",
        "Derivar: 1/(x·ln(a))"
      ],
      "stepByStep": [
        "📚 **Derivada de logaritmo con base arbitraria**",
        "",
        "**Fórmula:**",
        "$$\\frac{d}{dx}[\\log_a(x)] = \\frac{1}{x \\ln(a)}$$",
        "",
        "🧮 **Demostración (cambio de base)**",
        "",
        "**Paso 1:** Cambiar a logaritmo natural",
        "$$\\log_a(x) = \\frac{\\ln(x)}{\\ln(a)}$$",
        "",
        "**Paso 2:** Derivar",
        "$$\\frac{d}{dx}\\left[\\frac{\\ln(x)}{\\ln(a)}\\right]$$",
        "",
        "Como $\\ln(a)$ es constante:",
        "$$= \\frac{1}{\\ln(a)} \\cdot \\frac{d}{dx}[\\ln(x)]$$",
        "",
        "$$= \\frac{1}{\\ln(a)} \\cdot \\frac{1}{x}$$",
        "",
        "$$= \\frac{1}{x \\ln(a)}$$",
        "",
        "💡 **Casos especiales**",
        "",
        "**Base e (logaritmo natural):**",
        "$$\\frac{d}{dx}[\\log_e(x)] = \\frac{1}{x \\ln(e)} = \\frac{1}{x \\cdot 1} = \\frac{1}{x}$$ ✓",
        "",
        "**Base 10:**",
        "$$\\frac{d}{dx}[\\log_{10}(x)] = \\frac{1}{x \\ln(10)}$$",
        "",
        "**Base 2:**",
        "$$\\frac{d}{dx}[\\log_2(x)] = \\frac{1}{x \\ln(2)}$$",
        "",
        "✅ **Respuesta**",
        "$\\frac{1}{x \\ln(a)}$"
      ],
      "explanation": "d/dx[log_a(x)] = 1/(x·ln(a))"
    },
    {
      "id": "dl-006",
      "topic": "derivada-log-base-a",
      "question": "Calcula $\\frac{d}{dx}[\\log_{10}(x)]$",
      "type": "multiple-choice",
      "options": [
        "$\\frac{1}{x \\ln(10)}$",
        "$\\frac{\\ln(10)}{x}$",
        "$\\frac{1}{10x}$",
        "$\\frac{10}{x}$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Fórmula: d/dx[log_a(x)] = 1/(x·ln(a))",
        "Aquí a = 10",
        "= 1/(x·ln(10))"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\log_{10}(x)$**",
        "",
        "🧮 **Aplicar fórmula**",
        "",
        "Para $\\log_a(x)$:",
        "$$\\frac{d}{dx}[\\log_a(x)] = \\frac{1}{x \\ln(a)}$$",
        "",
        "Con $a = 10$:",
        "$$\\frac{d}{dx}[\\log_{10}(x)] = \\frac{1}{x \\ln(10)}$$",
        "",
        "📊 **Valor numérico**",
        "$$\\ln(10) \\approx 2.303$$",
        "",
        "Entonces:",
        "$$\\frac{d}{dx}[\\log_{10}(x)] \\approx \\frac{1}{2.303x}$$",
        "",
        "💡 **Conversión a ln(x)**",
        "",
        "$$\\log_{10}(x) = \\frac{\\ln(x)}{\\ln(10)}$$",
        "",
        "Derivando:",
        "$$\\frac{d}{dx}\\left[\\frac{\\ln(x)}{\\ln(10)}\\right] = \\frac{1}{\\ln(10)} \\cdot \\frac{1}{x} = \\frac{1}{x\\ln(10)}$$ ✓",
        "",
        "✅ **Respuesta**",
        "$\\frac{1}{x \\ln(10)}$"
      ],
      "explanation": "d/dx[log₁₀(x)] = 1/(x·ln(10))"
    },
    {
      "id": "dl-007",
      "topic": "derivada-log-base-a",
      "question": "Completa: $\\frac{d}{dx}[\\log_2(x)] = \\frac{1}{x \\cdot \\text{_____}}$",
      "type": "fill-blank",
      "blanks": ["ln(2)"],
      "distractors": ["log(2)", "2", "ln(x)", "e", "log₂(e)"],
      "template": "$\\frac{d}{dx}[\\log_2(x)] = \\frac{1}{x \\cdot \\text{_____}}$",
      "difficulty": "medio",
      "hints": [
        "Fórmula general: 1/(x·ln(a))",
        "Aquí a = 2",
        "Respuesta: ln(2)"
      ],
      "stepByStep": [
        "🎯 **Derivada de log₂(x)**",
        "",
        "📐 **Fórmula general**",
        "$$\\frac{d}{dx}[\\log_a(x)] = \\frac{1}{x \\ln(a)}$$",
        "",
        "🧮 **Con a = 2**",
        "$$\\frac{d}{dx}[\\log_2(x)] = \\frac{1}{x \\ln(2)}$$",
        "",
        "💡 **Valor numérico**",
        "$$\\ln(2) \\approx 0.693$$",
        "",
        "📊 **Relación con ln(x)**",
        "",
        "$$\\log_2(x) = \\frac{\\ln(x)}{\\ln(2)}$$",
        "",
        "✅ **Respuesta**",
        "$\\ln(2)$"
      ],
      "explanation": "d/dx[log₂(x)] = 1/(x·ln(2))"
    },
    {
      "id": "dl-008",
      "topic": "logaritmo-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\ln(3x)]$",
      "type": "multiple-choice",
      "options": [
        "$\\frac{1}{x}$",
        "$\\frac{3}{x}$",
        "$\\frac{1}{3x}$",
        "$3\\ln(x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "u = 3x, du/dx = 3",
        "d/dx[ln(u)] = (1/u)·du/dx",
        "= (1/3x)·3 = 1/x"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln(3x)$**",
        "",
        "🔗 **Método 1: Regla de la cadena**",
        "",
        "$$u = 3x, \\quad \\frac{du}{dx} = 3$$",
        "",
        "$$\\frac{d}{dx}[\\ln(u)] = \\frac{1}{u} \\cdot \\frac{du}{dx}$$",
        "",
        "$$= \\frac{1}{3x} \\cdot 3$$",
        "",
        "$$= \\frac{3}{3x} = \\frac{1}{x}$$",
        "",
        "📐 **Método 2: Propiedades de logaritmos**",
        "",
        "$$\\ln(3x) = \\ln(3) + \\ln(x)$$",
        "",
        "$$\\frac{d}{dx}[\\ln(3) + \\ln(x)] = 0 + \\frac{1}{x}$$",
        "",
        "$$= \\frac{1}{x}$$",
        "",
        "💡 **Patrón general**",
        "$$\\frac{d}{dx}[\\ln(kx)] = \\frac{1}{x}$$",
        "",
        "para cualquier constante $k > 0$.",
        "",
        "✅ **Respuesta**",
        "$\\frac{1}{x}$"
      ],
      "explanation": "d/dx[ln(3x)] = (1/3x)·3 = 1/x"
    },
    {
      "id": "dl-009",
      "topic": "logaritmo-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\ln(x^2)]$",
      "type": "multiple-choice",
      "options": [
        "$\\frac{2}{x}$",
        "$\\frac{1}{x^2}$",
        "$2\\ln(x)$",
        "$\\frac{2x}{x^2}$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "u = x², du/dx = 2x",
        "d/dx[ln(u)] = (1/u)·du/dx",
        "= (1/x²)·2x = 2/x"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln(x^2)$**",
        "",
        "🔗 **Método 1: Regla de la cadena**",
        "",
        "$$u = x^2, \\quad \\frac{du}{dx} = 2x$$",
        "",
        "$$\\frac{d}{dx}[\\ln(u)] = \\frac{1}{u} \\cdot \\frac{du}{dx}$$",
        "",
        "$$= \\frac{1}{x^2} \\cdot 2x$$",
        "",
        "$$= \\frac{2x}{x^2} = \\frac{2}{x}$$",
        "",
        "📐 **Método 2: Propiedad de logaritmos**",
        "",
        "$$\\ln(x^2) = 2\\ln(x)$$",
        "",
        "$$\\frac{d}{dx}[2\\ln(x)] = 2 \\cdot \\frac{1}{x}$$",
        "",
        "$$= \\frac{2}{x}$$",
        "",
        "💡 **Patrón general**",
        "$$\\frac{d}{dx}[\\ln(x^n)] = \\frac{n}{x}$$",
        "",
        "✅ **Respuesta**",
        "$\\frac{2}{x}$"
      ],
      "explanation": "d/dx[ln(x²)] = (1/x²)·2x = 2/x, o usando ln(x²) = 2ln(x)"
    },
    {
      "id": "dl-010",
      "topic": "logaritmo-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\ln(\\sin(x))]$",
      "type": "multiple-choice",
      "options": [
        "$\\cot(x)$",
        "$\\tan(x)$",
        "$\\frac{1}{\\sin(x)}$",
        "$\\cos(x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "u = sin(x), du/dx = cos(x)",
        "d/dx[ln(u)] = (1/u)·du/dx",
        "= cos(x)/sin(x) = cot(x)"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln(\\sin(x))$**",
        "",
        "🔗 **Regla de la cadena**",
        "",
        "**Funciones:**",
        "* Externa: $\\ln(u)$ con $u = \\sin(x)$",
        "* Interna: $\\sin(x)$",
        "",
        "📐 **Paso 1: Derivada de ln(u)**",
        "$$\\frac{d}{du}[\\ln(u)] = \\frac{1}{u}$$",
        "",
        "En $u = \\sin(x)$:",
        "$$\\frac{1}{\\sin(x)}$$",
        "",
        "🧮 **Paso 2: Derivada de sin(x)**",
        "$$\\frac{d}{dx}[\\sin(x)] = \\cos(x)$$",
        "",
        "🎯 **Paso 3: Multiplicar**",
        "$$\\frac{d}{dx}[\\ln(\\sin(x))] = \\frac{1}{\\sin(x)} \\cdot \\cos(x)$$",
        "",
        "$$= \\frac{\\cos(x)}{\\sin(x)}$$",
        "",
        "💡 **Simplificar usando identidades**",
        "$$\\frac{\\cos(x)}{\\sin(x)} = \\cot(x)$$",
        "",
        "✅ **Respuesta**",
        "$\\cot(x)$"
      ],
      "explanation": "d/dx[ln(sin(x))] = cos(x)/sin(x) = cot(x)"
    },
    {
      "id": "dl-011",
      "topic": "logaritmo-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\ln(e^x)]$",
      "type": "multiple-choice",
      "options": [
        "$1$",
        "$e^x$",
        "$\\frac{1}{e^x}$",
        "$x$"
      ],
      "correct": 0,
      "difficulty": "facil",
      "hints": [
        "Simplifica primero: ln(e^x) = x",
        "d/dx[x] = 1",
        "O por cadena: (1/e^x)·e^x = 1"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln(e^x)$**",
        "",
        "🎯 **Método 1: Simplificar primero**",
        "",
        "**Propiedad de logaritmos:**",
        "$$\\ln(e^x) = x \\ln(e) = x \\cdot 1 = x$$",
        "",
        "**Derivar:**",
        "$$\\frac{d}{dx}[x] = 1$$",
        "",
        "🔗 **Método 2: Regla de la cadena**",
        "",
        "$$u = e^x, \\quad \\frac{du}{dx} = e^x$$",
        "",
        "$$\\frac{d}{dx}[\\ln(u)] = \\frac{1}{u} \\cdot \\frac{du}{dx}$$",
        "",
        "$$= \\frac{1}{e^x} \\cdot e^x$$",
        "",
        "$$= 1$$",
        "",
        "💡 **Relación inversa**",
        "",
        "Como $\\ln(x)$ y $e^x$ son funciones inversas:",
        "$$\\ln(e^x) = x$$",
        "$$e^{\\ln(x)} = x$$",
        "",
        "✅ **Respuesta**",
        "$1$"
      ],
      "explanation": "ln(e^x) = x, entonces d/dx[x] = 1"
    },
    {
      "id": "dl-012",
      "topic": "logaritmo-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\ln(\\sqrt{x})]$",
      "type": "multiple-choice",
      "options": [
        "$\\frac{1}{2x}$",
        "$\\frac{2}{x}$",
        "$\\frac{1}{x}$",
        "$\\frac{1}{\\sqrt{x}}$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "√x = x^(1/2)",
        "ln(x^(1/2)) = (1/2)ln(x)",
        "d/dx[(1/2)ln(x)] = 1/(2x)"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln(\\sqrt{x})$**",
        "",
        "📐 **Método 1: Propiedad de logaritmos**",
        "",
        "$$\\sqrt{x} = x^{1/2}$$",
        "",
        "$$\\ln(x^{1/2}) = \\frac{1}{2}\\ln(x)$$",
        "",
        "**Derivar:**",
        "$$\\frac{d}{dx}\\left[\\frac{1}{2}\\ln(x)\\right] = \\frac{1}{2} \\cdot \\frac{1}{x}$$",
        "",
        "$$= \\frac{1}{2x}$$",
        "",
        "🔗 **Método 2: Regla de la cadena**",
        "",
        "$$u = \\sqrt{x} = x^{1/2}, \\quad \\frac{du}{dx} = \\frac{1}{2}x^{-1/2} = \\frac{1}{2\\sqrt{x}}$$",
        "",
        "$$\\frac{d}{dx}[\\ln(u)] = \\frac{1}{u} \\cdot \\frac{du}{dx}$$",
        "",
        "$$= \\frac{1}{\\sqrt{x}} \\cdot \\frac{1}{2\\sqrt{x}}$$",
        "",
        "$$= \\frac{1}{2x}$$",
        "",
        "✅ **Respuesta**",
        "$\\frac{1}{2x}$"
      ],
      "explanation": "ln(√x) = (1/2)ln(x), d/dx = 1/(2x)"
    },
    {
      "id": "dl-013",
      "topic": "propiedades-logaritmicas",
      "question": "Simplifica antes de derivar: $\\frac{d}{dx}[\\ln(x^3 \\cdot e^{2x})]$",
      "type": "multiple-choice",
      "options": [
        "$\\frac{3}{x} + 2$",
        "$\\frac{3 + 2x}{x}$",
        "$\\frac{5}{x}$",
        "$3\\ln(x) + 2x$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "ln(ab) = ln(a) + ln(b)",
        "ln(x³·e^(2x)) = 3ln(x) + 2x",
        "Derivar: 3/x + 2"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln(x^3 \\cdot e^{2x})$**",
        "",
        "🧮 **Paso 1: Aplicar propiedades**",
        "",
        "**Propiedad del producto:**",
        "$$\\ln(ab) = \\ln(a) + \\ln(b)$$",
        "",
        "$$\\ln(x^3 \\cdot e^{2x}) = \\ln(x^3) + \\ln(e^{2x})$$",
        "",
        "📐 **Paso 2: Simplificar cada término**",
        "",
        "**Primer término:**",
        "$$\\ln(x^3) = 3\\ln(x)$$",
        "",
        "**Segundo término:**",
        "$$\\ln(e^{2x}) = 2x \\ln(e) = 2x \\cdot 1 = 2x$$",
        "",
        "**Resultado:**",
        "$$\\ln(x^3 \\cdot e^{2x}) = 3\\ln(x) + 2x$$",
        "",
        "🎯 **Paso 3: Derivar**",
        "$$\\frac{d}{dx}[3\\ln(x) + 2x]$$",
        "",
        "$$= 3 \\cdot \\frac{1}{x} + 2$$",
        "",
        "$$= \\frac{3}{x} + 2$$",
        "",
        "💡 **Ventaja de simplificar**",
        "",
        "Sin simplificar, necesitarías regla de la cadena compleja.",
        "Con propiedades, ¡solo derivadas simples!",
        "",
        "✅ **Respuesta**",
        "$\\frac{3}{x} + 2$"
      ],
      "explanation": "ln(x³·e^(2x)) = 3ln(x) + 2x, derivar: 3/x + 2"
    },
    {
      "id": "dl-014",
      "topic": "propiedades-logaritmicas",
      "question": "Calcula $\\frac{d}{dx}\\left[\\ln\\left(\\frac{x^2}{e^x}\\right)\\right]$ usando propiedades",
      "type": "multiple-choice",
      "options": [
        "$\\frac{2}{x} - 1$",
        "$\\frac{2 - x}{x}$",
        "$\\frac{2}{x^2 e^x}$",
        "$2\\ln(x) - e^x$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "ln(a/b) = ln(a) - ln(b)",
        "ln(x²/e^x) = 2ln(x) - x",
        "Derivar: 2/x - 1"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln\\left(\\frac{x^2}{e^x}\\right)$**",
        "",
        "🧮 **Paso 1: Propiedad del cociente**",
        "",
        "$$\\ln\\left(\\frac{a}{b}\\right) = \\ln(a) - \\ln(b)$$",
        "",
        "$$\\ln\\left(\\frac{x^2}{e^x}\\right) = \\ln(x^2) - \\ln(e^x)$$",
        "",
        "📐 **Paso 2: Simplificar**",
        "",
        "**Primer término:**",
        "$$\\ln(x^2) = 2\\ln(x)$$",
        "",
        "**Segundo término:**",
        "$$\\ln(e^x) = x$$",
        "",
        "**Resultado:**",
        "$$\\ln\\left(\\frac{x^2}{e^x}\\right) = 2\\ln(x) - x$$",
        "",
        "🎯 **Paso 3: Derivar**",
        "$$\\frac{d}{dx}[2\\ln(x) - x]$$",
        "",
        "$$= 2 \\cdot \\frac{1}{x} - 1$$",
        "",
        "$$= \\frac{2}{x} - 1$$",
        "",
        "💡 **Forma alternativa**",
        "$$\\frac{2}{x} - 1 = \\frac{2 - x}{x}$$",
        "",
        "✅ **Respuesta**",
        "$\\frac{2}{x} - 1$"
      ],
      "explanation": "ln(x²/e^x) = 2ln(x) - x, d/dx = 2/x - 1"
    },
    {
      "id": "dl-015",
      "topic": "propiedades-logaritmicas",
      "question": "Ordena las propiedades de logaritmos por orden de aplicación para simplificar $\\ln\\left(\\frac{x^3 \\sqrt{y}}{z^2}\\right)$",
      "type": "ordering",
      "items": [
        "Aplicar propiedad del cociente: $\\ln(a/b) = \\ln(a) - \\ln(b)$",
        "Aplicar propiedad del producto: $\\ln(ab) = \\ln(a) + \\ln(b)$",
        "Aplicar propiedad de la potencia: $\\ln(x^n) = n\\ln(x)$",
        "Resultado final: $3\\ln(x) + \\frac{1}{2}\\ln(y) - 2\\ln(z)$"
      ],
      "correctOrder": [0, 1, 2, 3],
      "difficulty": "medio",
      "hints": [
        "Primero: separar numerador y denominador",
        "Segundo: separar factores en numerador",
        "Tercero: bajar exponentes",
        "Cuarto: expresión simplificada"
      ],
      "stepByStep": [
        "🔗 **Simplificación: $\\ln\\left(\\frac{x^3 \\sqrt{y}}{z^2}\\right)$**",
        "",
        "**Paso 1: Propiedad del cociente**",
        "$$\\ln\\left(\\frac{x^3 \\sqrt{y}}{z^2}\\right) = \\ln(x^3 \\sqrt{y}) - \\ln(z^2)$$",
        "",
        "**Paso 2: Propiedad del producto**",
        "$$= \\ln(x^3) + \\ln(\\sqrt{y}) - \\ln(z^2)$$",
        "",
        "**Paso 3: Propiedad de la potencia**",
        "",
        "$\\ln(x^3) = 3\\ln(x)$",
        "",
        "$\\ln(\\sqrt{y}) = \\ln(y^{1/2}) = \\frac{1}{2}\\ln(y)$",
        "",
        "$\\ln(z^2) = 2\\ln(z)$",
        "",
        "**Paso 4: Resultado**",
        "$$= 3\\ln(x) + \\frac{1}{2}\\ln(y) - 2\\ln(z)$$",
        "",
        "💡 **Derivar ahora es fácil**",
        "$$\\frac{d}{dx}\\left[3\\ln(x) + \\frac{1}{2}\\ln(y) - 2\\ln(z)\\right]$$",
        "",
        "(Tratando y, z como constantes respecto a x)",
        "",
        "$$= \\frac{3}{x}$$",
        "",
        "✅ **Orden correcto**",
        "Cociente → Producto → Potencia → Resultado"
      ],
      "explanation": "Cociente → Producto → Potencia → Resultado simplificado"
    },
    {
      "id": "dl-016",
      "topic": "propiedades-logaritmicas",
      "question": "Clasifica las expresiones según si se debe usar propiedades antes de derivar o regla de la cadena directa",
      "description": "Organiza según la estrategia más eficiente.",
      "type": "categorize",
      "items": [
        "$\\ln(x^5)$",
        "$\\ln(3x + 1)$",
        "$\\ln(x \\cdot e^x)$",
        "$\\ln(\\sin^2(x))$",
        "$\\ln\\left(\\frac{x^2}{x+1}\\right)$"
      ],
      "categories": {
        "propiedades": "Usar propiedades primero",
        "cadena-directa": "Regla de la cadena directamente",
        "ambas": "Puede usarse cualquiera"
      },
      "correctCategories": {
        "$\\ln(x^5)$": "propiedades",
        "$\\ln(3x + 1)$": "cadena-directa",
        "$\\ln(x \\cdot e^x)$": "propiedades",
        "$\\ln(\\sin^2(x))$": "ambas",
        "$\\ln\\left(\\frac{x^2}{x+1}\\right)$": "ambas"
      },
      "difficulty": "medio",
      "hints": [
        "Potencias simples: usa propiedades",
        "Funciones compuestas: cadena directa",
        "Productos/cocientes: propiedades primero"
      ],
      "stepByStep": [
        "📊 **Clasificación de estrategias**",
        "",
        "**USAR PROPIEDADES PRIMERO:**",
        "",
        "1. **$\\ln(x^5)$**",
        "   - Propiedad: $\\ln(x^5) = 5\\ln(x)$",
        "   - Derivar: $5/x$ (simple)",
        "   - Cadena directa: $(1/x^5) \\cdot 5x^4 = 5/x$ (complejo)",
        "",
        "2. **$\\ln(x \\cdot e^x)$**",
        "   - Propiedad: $\\ln(x) + \\ln(e^x) = \\ln(x) + x$",
        "   - Derivar: $1/x + 1$ (simple)",
        "",
        "**CADENA DIRECTA:**",
        "",
        "3. **$\\ln(3x + 1)$**",
        "   - No hay propiedades aplicables",
        "   - Cadena: $(1/(3x+1)) \\cdot 3 = 3/(3x+1)$",
        "",
        "**AMBAS (preferir propiedades):**",
        "",
        "4. **$\\ln(\\sin^2(x))$**",
        "   - Propiedad: $2\\ln(\\sin(x))$, derivar: $2\\cot(x)$",
        "   - Cadena: $(1/\\sin^2(x)) \\cdot 2\\sin(x)\\cos(x) = 2\\cot(x)$",
        "",
        "5. **$\\ln\\left(\\frac{x^2}{x+1}\\right)$**",
        "   - Propiedad: $2\\ln(x) - \\ln(x+1)$",
        "   - O regla del cociente en cadena",
        "",
        "💡 **Recomendación**",
        "Siempre busca simplificar con propiedades antes de derivar.",
        "",
        "✅ **Clasificación**",
        "* Propiedades: x^5, x·e^x",
        "* Cadena directa: 3x+1",
        "* Ambas: sin²(x), x²/(x+1)"
      ],
      "explanation": "Propiedades: potencias simples, productos. Cadena: composiciones sin propiedades"
    },
    {
      "id": "dl-017",
      "topic": "aplicaciones-logaritmicas",
      "question": "La escala de pH se define como $pH = -\\log_{10}[H^+]$. Si $[H^+] = 10^{-7}$ M (neutro), ¿cuál es el pH? Ingresa tu respuesta como número entero.",
      "type": "numeric",
      "correct": 7,
      "tolerance": 0.1,
      "difficulty": "medio",
      "instructions": "Calcula pH = -log₁₀(10⁻⁷) = -(-7) = 7. Ingresa el valor entero 7 (no 7.0 ni 7.00).",
      "format": "integer",
      "examples": ["Respuesta: 7 (no 7.0 ni 7.00)"],
      "hints": [
        "pH = -log₁₀(10^(-7))",
        "log₁₀(10^(-7)) = -7",
        "pH = -(-7) = 7"
      ],
      "stepByStep": [
        "📝 **Calcular pH**",
        "",
        "**Fórmula:**",
        "$$pH = -\\log_{10}[H^+]$$",
        "",
        "**Dato:**",
        "$$[H^+] = 10^{-7} \\text{ M}$$",
        "",
        "🧮 **Sustituir**",
        "$$pH = -\\log_{10}(10^{-7})$$",
        "",
        "📐 **Simplificar**",
        "",
        "Propiedad: $\\log_a(a^x) = x$",
        "",
        "$$\\log_{10}(10^{-7}) = -7$$",
        "",
        "Entonces:",
        "$$pH = -(-7) = 7$$",
        "",
        "💡 **Interpretación**",
        "",
        "**pH = 7:** Neutro (agua pura)",
        "**pH < 7:** Ácido",
        "**pH > 7:** Básico",
        "",
        "📊 **Relación con derivada**",
        "",
        "$$\\frac{dpH}{d[H^+]} = -\\frac{1}{[H^+] \\ln(10)}$$",
        "",
        "Tasa de cambio del pH respecto a concentración.",
        "",
        "✅ **Respuesta**",
        "pH = **7**"
      ],
      "explanation": "pH = -log₁₀(10^(-7)) = -(-7) = 7"
    },
    {
      "id": "dl-018",
      "topic": "aplicaciones-logaritmicas",
      "question": "La escala de decibeles es $dB = 10\\log_{10}\\left(\\frac{I}{I_0}\\right)$. ¿Cuál es la derivada $\\frac{d(dB)}{dI}$?",
      "type": "multiple-choice",
      "options": [
        "$\\frac{10}{I \\ln(10)}$",
        "$\\frac{10}{I}$",
        "$\\frac{\\ln(10)}{I}$",
        "$\\frac{I_0}{I}$"
      ],
      "correct": 0,
      "difficulty": "avanzado",
      "hints": [
        "dB = 10·log₁₀(I) - 10·log₁₀(I₀)",
        "I₀ es constante",
        "d/dI[10·log₁₀(I)] = 10/(I·ln(10))"
      ],
      "stepByStep": [
        "📝 **Derivar escala de decibeles**",
        "",
        "**Fórmula:**",
        "$$dB = 10\\log_{10}\\left(\\frac{I}{I_0}\\right)$$",
        "",
        "🧮 **Paso 1: Aplicar propiedad**",
        "",
        "$$\\log_{10}\\left(\\frac{I}{I_0}\\right) = \\log_{10}(I) - \\log_{10}(I_0)$$",
        "",
        "Entonces:",
        "$$dB = 10\\log_{10}(I) - 10\\log_{10}(I_0)$$",
        "",
        "📐 **Paso 2: Derivar respecto a I**",
        "",
        "$I_0$ es constante:",
        "",
        "$$\\frac{d(dB)}{dI} = 10 \\cdot \\frac{d}{dI}[\\log_{10}(I)] - 0$$",
        "",
        "🎯 **Paso 3: Aplicar fórmula**",
        "",
        "$$\\frac{d}{dI}[\\log_{10}(I)] = \\frac{1}{I \\ln(10)}$$",
        "",
        "Entonces:",
        "$$\\frac{d(dB)}{dI} = 10 \\cdot \\frac{1}{I \\ln(10)}$$",
        "",
        "$$= \\frac{10}{I \\ln(10)}$$",
        "",
        "💡 **Interpretación**",
        "",
        "Tasa de cambio de decibeles respecto a intensidad.",
        "",
        "Inversamente proporcional a $I$: cambios pequeños en $I$ bajo → cambios grandes en dB.",
        "",
        "✅ **Respuesta**",
        "$\\frac{10}{I \\ln(10)}$"
      ],
      "explanation": "d(dB)/dI = 10·d/dI[log₁₀(I)] = 10/(I·ln(10))"
    },
    {
      "id": "dl-019",
      "topic": "aplicaciones-logaritmicas",
      "question": "En economía, la elasticidad ingreso se define como $E = \\frac{d\\ln(Q)}{d\\ln(I)}$. Si $Q = I^{0.8}$, ¿cuál es $E$? Ingresa tu respuesta como decimal con una cifra decimal.",
      "type": "numeric",
      "correct": 0.8,
      "tolerance": 0.05,
      "difficulty": "avanzado",
      "instructions": "Calcula ln(Q) = 0.8·ln(I), entonces d[ln(Q)]/d[ln(I)] = 0.8. Ingresa 0.8",
      "format": "decimal_1_place",
      "acceptedAlternatives": ["4/5"],
      "hints": [
        "Q = I^0.8, entonces ln(Q) = 0.8·ln(I)",
        "d[ln(Q)]/d[ln(I)] = 0.8",
        "E = 0.8"
      ],
      "stepByStep": [
        "📝 **Calcular elasticidad ingreso**",
        "",
        "**Fórmula:**",
        "$$E = \\frac{d\\ln(Q)}{d\\ln(I)}$$",
        "",
        "**Dato:**",
        "$$Q = I^{0.8}$$",
        "",
        "🧮 **Paso 1: Aplicar logaritmo**",
        "",
        "$$\\ln(Q) = \\ln(I^{0.8})$$",
        "",
        "$$\\ln(Q) = 0.8 \\ln(I)$$",
        "",
        "📐 **Paso 2: Derivar**",
        "",
        "Sea $u = \\ln(I)$, entonces $\\ln(Q) = 0.8u$",
        "",
        "$$\\frac{d\\ln(Q)}{du} = 0.8$$",
        "",
        "Como $u = \\ln(I)$:",
        "$$\\frac{d\\ln(Q)}{d\\ln(I)} = 0.8$$",
        "",
        "🎯 **Resultado**",
        "$$E = 0.8$$",
        "",
        "💡 **Interpretación económica**",
        "",
        "**E = 0.8:** Bien normal (0 < E < 1)",
        "",
        "Un aumento del 10% en ingreso ($I$) causa un aumento del 8% en cantidad demandada ($Q$).",
        "",
        "**Clasificación:**",
        "* E > 1: Bien de lujo",
        "- 0 < E < 1: Bien normal",
        "* E < 0: Bien inferior",
        "",
        "✅ **Respuesta**",
        "E = **0.8**"
      ],
      "explanation": "ln(Q) = 0.8·ln(I), entonces d[ln(Q)]/d[ln(I)] = 0.8"
    },
    {
      "id": "dl-020",
      "topic": "aplicaciones-logaritmicas",
      "question": "¿Cuál es la derivada de $f(x) = x\\ln(x)$ (producto)?",
      "type": "multiple-choice",
      "options": [
        "$\\ln(x) + 1$",
        "$\\frac{1}{x}$",
        "$x\\left(\\ln(x) + \\frac{1}{x}\\right)$",
        "$\\frac{x}{\\ln(x)}$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Regla del producto: (uv)' = u'v + uv'",
        "u = x, u' = 1",
        "v = ln(x), v' = 1/x"
      ],
      "stepByStep": [
        "📝 **Derivar: $x\\ln(x)$**",
        "",
        "🔗 **Regla del producto**",
        "",
        "$(uv)' = u'v + uv'$",
        "",
        "🧮 **Identificar**",
        "* $1 = x$, entonces $u' = 1$",
        "* $1 = \\ln(x)$, entonces $v' = \\frac{1}{x}$",
        "",
        "📐 **Aplicar**",
        "$$f'(x) = u'v + uv'$$",
        "",
        "$$= (1)(\\ln(x)) + (x)\\left(\\frac{1}{x}\\right)$$",
        "",
        "$$= \\ln(x) + 1$$",
        "",
        "💡 **Verificación**",
        "También se puede escribir:",
        "$$f'(x) = 1 + \\ln(x)$$",
        "",
        "📊 **Punto crítico**",
        "",
        "**f'(x) = 0 cuando:**",
        "$$\\ln(x) + 1 = 0$$",
        "",
        "$$\\ln(x) = -1$$",
        "",
        "$$x = e^{-1} = \\frac{1}{e}$$",
        "",
        "✅ **Respuesta**",
        "$\\ln(x) + 1$"
      ],
      "explanation": "Por producto: 1·ln(x) + x·(1/x) = ln(x) + 1"
    },
    {
      "id": "dl-021",
      "topic": "aplicaciones-logaritmicas",
      "question": "¿Cuál es la derivada de $g(x) = \\frac{\\ln(x)}{x}$ (cociente)?",
      "type": "multiple-choice",
      "options": [
        "$\\frac{1 - \\ln(x)}{x^2}$",
        "$\\frac{\\ln(x) - 1}{x^2}$",
        "$\\frac{1}{x^2}$",
        "$\\frac{1}{x\\ln(x)}$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Regla del cociente: (u/v)' = (u'v - uv')/v²",
        "u = ln(x), u' = 1/x",
        "v = x, v' = 1"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\frac{\\ln(x)}{x}$**",
        "",
        "🔗 **Regla del cociente**",
        "",
        "$$\\left(\\frac{u}{v}\\right)' = \\frac{u'v - uv'}{v^2}$$",
        "",
        "🧮 **Identificar**",
        "* $1 = \\ln(x)$, entonces $u' = \\frac{1}{x}$",
        "* $1 = x$, entonces $v' = 1$",
        "",
        "📐 **Aplicar**",
        "$$g'(x) = \\frac{\\frac{1}{x} \\cdot x - \\ln(x) \\cdot 1}{x^2}$$",
        "",
        "$$= \\frac{1 - \\ln(x)}{x^2}$$",
        "",
        "💡 **Punto crítico**",
        "",
        "**g'(x) = 0 cuando:**",
        "$$1 - \\ln(x) = 0$$",
        "",
        "$$\\ln(x) = 1$$",
        "",
        "$$x = e$$",
        "",
        "📊 **Análisis**",
        "",
        "* $1 < e$: $g'(x) > 0$ (creciente)",
        "* $1 = e$: máximo local",
        "* $1 > e$: $g'(x) < 0$ (decreciente)",
        "",
        "✅ **Respuesta**",
        "$\\frac{1 - \\ln(x)}{x^2}$"
      ],
      "explanation": "Por cociente: [(1/x)·x - ln(x)·1]/x² = (1 - ln(x))/x²"
    },
    {
      "id": "dl-022",
      "topic": "aplicaciones-logaritmicas",
      "question": "Si $y = \\ln(x^x)$, ¿cuál es $\\frac{dy}{dx}$ (usa propiedades)?",
      "type": "multiple-choice",
      "options": [
        "$1 + \\ln(x)$",
        "$\\frac{1}{x}$",
        "$x\\ln(x)$",
        "$\\frac{x}{\\ln(x)}$"
      ],
      "correct": 0,
      "difficulty": "avanzado",
      "hints": [
        "ln(x^x) = x·ln(x) (propiedad)",
        "Derivar producto: (x·ln(x))'",
        "= ln(x) + 1"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\ln(x^x)$**",
        "",
        "🧮 **Paso 1: Simplificar con propiedades**",
        "",
        "**Propiedad de potencia:**",
        "$$\\ln(x^x) = x \\ln(x)$$",
        "",
        "📐 **Paso 2: Derivar (producto)**",
        "",
        "$$\\frac{dy}{dx} = \\frac{d}{dx}[x \\ln(x)]$$",
        "",
        "**Regla del producto:**",
        "$$= (1)(\\ln(x)) + (x)\\left(\\frac{1}{x}\\right)$$",
        "",
        "$$= \\ln(x) + 1$$",
        "",
        "🎯 **Forma alternativa**",
        "$$\\frac{dy}{dx} = 1 + \\ln(x)$$",
        "",
        "💡 **Nota importante**",
        "",
        "$x^x$ NO es lo mismo que $x \\cdot x = x^2$.",
        "",
        "$x^x$ significa \"x elevado a la potencia x\" (varía el exponente).",
        "",
        "📊 **Para derivar x^x directamente**",
        "",
        "$$\\frac{d}{dx}[x^x] = x^x(1 + \\ln(x))$$",
        "",
        "(Requiere derivación logarítmica avanzada)",
        "",
        "✅ **Respuesta**",
        "$1 + \\ln(x)$"
      ],
      "explanation": "ln(x^x) = x·ln(x), derivar: ln(x) + 1"
    }
  ]
}