{
  "derivadas-trigonometricas": [
    {
      "id": "dt-001",
      "topic": "derivada-seno-coseno",
      "question": "¿Cuál es la derivada de $f(x) = \\sin(x)$?",
      "type": "multiple-choice",
      "options": [
        "$\\cos(x)$",
        "$-\\sin(x)$",
        "$-\\cos(x)$",
        "$\\tan(x)$"
      ],
      "correct": 0,
      "difficulty": "facil",
      "hints": [
        "Es una de las derivadas fundamentales",
        "La derivada de sin(x) es cos(x)",
        "Memoriza: d/dx[sin(x)] = cos(x)"
      ],
      "stepByStep": [
        "📚 **Derivada del seno**",
        "",
        "**Fórmula fundamental:**",
        "$$\\frac{d}{dx}[\\sin(x)] = \\cos(x)$$",
        "",
        "💡 **Demostración (por límite)**",
        "",
        "$$\\frac{d}{dx}[\\sin(x)] = \\lim_{h \\to 0} \\frac{\\sin(x+h) - \\sin(x)}{h}$$",
        "",
        "Usando identidades trigonométricas:",
        "$$= \\lim_{h \\to 0} \\frac{\\sin(x)\\cos(h) + \\cos(x)\\sin(h) - \\sin(x)}{h}$$",
        "",
        "$$= \\sin(x) \\lim_{h \\to 0} \\frac{\\cos(h) - 1}{h} + \\cos(x) \\lim_{h \\to 0} \\frac{\\sin(h)}{h}$$",
        "",
        "Como $\\lim_{h \\to 0} \\frac{\\cos(h) - 1}{h} = 0$ y $\\lim_{h \\to 0} \\frac{\\sin(h)}{h} = 1$:",
        "",
        "$$= \\sin(x) \\cdot 0 + \\cos(x) \\cdot 1 = \\cos(x)$$",
        "",
        "✅ **Respuesta**",
        "$\\frac{d}{dx}[\\sin(x)] = \\cos(x)$"
      ],
      "explanation": "La derivada de sin(x) es cos(x), una fórmula fundamental del cálculo"
    },
    {
      "id": "dt-002",
      "topic": "derivada-seno-coseno",
      "question": "¿Cuál es la derivada de $f(x) = \\cos(x)$?",
      "type": "multiple-choice",
      "options": [
        "$-\\sin(x)$",
        "$\\sin(x)$",
        "$-\\cos(x)$",
        "$\\cos(x)$"
      ],
      "correct": 0,
      "difficulty": "facil",
      "hints": [
        "Tiene signo negativo",
        "d/dx[cos(x)] = -sin(x)",
        "El coseno deriva a menos seno"
      ],
      "stepByStep": [
        "📚 **Derivada del coseno**",
        "",
        "**Fórmula fundamental:**",
        "$$\\frac{d}{dx}[\\cos(x)] = -\\sin(x)$$",
        "",
        "💡 **Nota importante**",
        "¡Observa el signo **negativo**!",
        "",
        "🧮 **Demostración (por límite)**",
        "",
        "$$\\frac{d}{dx}[\\cos(x)] = \\lim_{h \\to 0} \\frac{\\cos(x+h) - \\cos(x)}{h}$$",
        "",
        "Usando identidades:",
        "$$= \\lim_{h \\to 0} \\frac{\\cos(x)\\cos(h) - \\sin(x)\\sin(h) - \\cos(x)}{h}$$",
        "",
        "$$= \\cos(x) \\lim_{h \\to 0} \\frac{\\cos(h) - 1}{h} - \\sin(x) \\lim_{h \\to 0} \\frac{\\sin(h)}{h}$$",
        "",
        "$$= \\cos(x) \\cdot 0 - \\sin(x) \\cdot 1 = -\\sin(x)$$",
        "",
        "📊 **Patrón cíclico**",
        "- $\\sin(x) \\to \\cos(x) \\to -\\sin(x) \\to -\\cos(x) \\to \\sin(x)$",
        "",
        "✅ **Respuesta**",
        "$\\frac{d}{dx}[\\cos(x)] = -\\sin(x)$"
      ],
      "explanation": "La derivada de cos(x) es -sin(x), nota el signo negativo"
    },
    {
      "id": "dt-003",
      "topic": "derivada-seno-coseno",
      "question": "Completa la tabla de derivadas trigonométricas básicas",
      "description": "Relaciona cada función con su derivada.",
      "type": "drag-drop",
      "items": [
        "$\\sin(x)$",
        "$\\cos(x)$",
        "$-\\sin(x)$",
        "$-\\cos(x)$"
      ],
      "categories": [
        "Derivada de $\\sin(x)$",
        "Derivada de $\\cos(x)$",
        "Derivada de $-\\cos(x)$",
        "Derivada de $-\\sin(x)$"
      ],
      "correctMapping": [1, 0, 3, 2],
      "difficulty": "facil",
      "hints": [
        "d/dx[sin(x)] = cos(x)",
        "d/dx[cos(x)] = -sin(x)",
        "Usa regla de la constante multiplicativa"
      ],
      "stepByStep": [
        "📋 **Tabla de derivadas sin y cos**",
        "",
        "**Fórmulas básicas:**",
        "",
        "1. $\\frac{d}{dx}[\\sin(x)] = \\cos(x)$",
        "",
        "2. $\\frac{d}{dx}[\\cos(x)] = -\\sin(x)$",
        "",
        "**Con constante negativa:**",
        "",
        "3. $\\frac{d}{dx}[-\\cos(x)] = -(-\\sin(x)) = \\sin(x)$",
        "",
        "4. $\\frac{d}{dx}[-\\sin(x)] = -\\cos(x)$",
        "",
        "🔄 **Patrón cíclico completo**",
        "",
        "Derivando sucesivamente $\\sin(x)$:",
        "$$\\sin(x) \\xrightarrow{d/dx} \\cos(x) \\xrightarrow{d/dx} -\\sin(x) \\xrightarrow{d/dx} -\\cos(x) \\xrightarrow{d/dx} \\sin(x)$$",
        "",
        "¡Ciclo de 4 derivadas!",
        "",
        "✅ **Respuestas**",
        "* sin(x) → cos(x)",
        "* cos(x) → -sin(x)",
        "- -cos(x) → sin(x)",
        "- -sin(x) → -cos(x)"
      ],
      "explanation": "sin→cos, cos→-sin, -cos→sin, -sin→-cos"
    },
    {
      "id": "dt-004",
      "topic": "derivada-seno-coseno",
      "question": "Calcula $\\frac{d}{dx}[3\\sin(x) - 2\\cos(x)]$",
      "type": "multiple-choice",
      "options": [
        "$3\\cos(x) + 2\\sin(x)$",
        "$3\\cos(x) - 2\\sin(x)$",
        "$-3\\cos(x) + 2\\sin(x)$",
        "$-3\\cos(x) - 2\\sin(x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Deriva término por término",
        "d/dx[3sin(x)] = 3cos(x)",
        "d/dx[-2cos(x)] = -2(-sin(x)) = 2sin(x)"
      ],
      "stepByStep": [
        "📝 **Derivar: $3\\sin(x) - 2\\cos(x)$**",
        "",
        "🧮 **Paso 1: Separar términos**",
        "$$\\frac{d}{dx}[3\\sin(x) - 2\\cos(x)] = \\frac{d}{dx}[3\\sin(x)] - \\frac{d}{dx}[2\\cos(x)]$$",
        "",
        "📐 **Paso 2: Derivar primer término**",
        "$$\\frac{d}{dx}[3\\sin(x)] = 3 \\cdot \\frac{d}{dx}[\\sin(x)] = 3\\cos(x)$$",
        "",
        "🎯 **Paso 3: Derivar segundo término**",
        "$$\\frac{d}{dx}[2\\cos(x)] = 2 \\cdot \\frac{d}{dx}[\\cos(x)] = 2(-\\sin(x)) = -2\\sin(x)$$",
        "",
        "📊 **Paso 4: Combinar**",
        "$$3\\cos(x) - (-2\\sin(x))$$",
        "$$= 3\\cos(x) + 2\\sin(x)$$",
        "",
        "✅ **Respuesta**",
        "$3\\cos(x) + 2\\sin(x)$"
      ],
      "explanation": "Deriva cada término: 3sin(x)→3cos(x), -2cos(x)→2sin(x)"
    },
    {
      "id": "dt-005",
      "topic": "derivada-tangente-cotangente",
      "question": "¿Cuál es la derivada de $f(x) = \\tan(x)$?",
      "type": "multiple-choice",
      "options": [
        "$\\sec^2(x)$",
        "$\\tan^2(x)$",
        "$-\\csc^2(x)$",
        "$\\cos^2(x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Usa tan(x) = sin(x)/cos(x)",
        "Aplica regla del cociente",
        "Resultado: sec²(x)"
      ],
      "stepByStep": [
        "📚 **Derivada de la tangente**",
        "",
        "**Fórmula:**",
        "$$\\frac{d}{dx}[\\tan(x)] = \\sec^2(x)$$",
        "",
        "🧮 **Demostración**",
        "",
        "**Método 1: Regla del cociente**",
        "",
        "$$\\tan(x) = \\frac{\\sin(x)}{\\cos(x)}$$",
        "",
        "$$\\frac{d}{dx}[\\tan(x)] = \\frac{\\cos(x) \\cdot \\cos(x) - \\sin(x) \\cdot (-\\sin(x))}{\\cos^2(x)}$$",
        "",
        "$$= \\frac{\\cos^2(x) + \\sin^2(x)}{\\cos^2(x)}$$",
        "",
        "Usando identidad $\\sin^2(x) + \\cos^2(x) = 1$:",
        "",
        "$$= \\frac{1}{\\cos^2(x)} = \\sec^2(x)$$",
        "",
        "💡 **Forma alternativa**",
        "$$\\frac{d}{dx}[\\tan(x)] = 1 + \\tan^2(x)$$",
        "",
        "(usando identidad $\\sec^2(x) = 1 + \\tan^2(x)$)",
        "",
        "✅ **Respuesta**",
        "$\\sec^2(x)$"
      ],
      "explanation": "d/dx[tan(x)] = sec²(x), derivable usando regla del cociente"
    },
    {
      "id": "dt-006",
      "topic": "derivada-tangente-cotangente",
      "question": "¿Cuál es la derivada de $f(x) = \\cot(x)$?",
      "type": "multiple-choice",
      "options": [
        "$-\\csc^2(x)$",
        "$\\csc^2(x)$",
        "$-\\sec^2(x)$",
        "$\\sec^2(x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "cot(x) = cos(x)/sin(x)",
        "Usa regla del cociente",
        "Resultado tiene signo negativo"
      ],
      "stepByStep": [
        "📚 **Derivada de la cotangente**",
        "",
        "**Fórmula:**",
        "$$\\frac{d}{dx}[\\cot(x)] = -\\csc^2(x)$$",
        "",
        "🧮 **Demostración**",
        "",
        "$$\\cot(x) = \\frac{\\cos(x)}{\\sin(x)}$$",
        "",
        "Regla del cociente:",
        "$$\\frac{d}{dx}[\\cot(x)] = \\frac{\\sin(x) \\cdot (-\\sin(x)) - \\cos(x) \\cdot \\cos(x)}{\\sin^2(x)}$$",
        "",
        "$$= \\frac{-\\sin^2(x) - \\cos^2(x)}{\\sin^2(x)}$$",
        "",
        "$$= \\frac{-(\\sin^2(x) + \\cos^2(x))}{\\sin^2(x)}$$",
        "",
        "$$= \\frac{-1}{\\sin^2(x)} = -\\csc^2(x)$$",
        "",
        "💡 **Nota del signo negativo**",
        "¡Siempre negativo!",
        "",
        "📊 **Forma alternativa**",
        "$$\\frac{d}{dx}[\\cot(x)] = -(1 + \\cot^2(x))$$",
        "",
        "✅ **Respuesta**",
        "$-\\csc^2(x)$"
      ],
      "explanation": "d/dx[cot(x)] = -csc²(x), nota el signo negativo"
    },
    {
      "id": "dt-007",
      "topic": "derivada-secante-cosecante",
      "question": "¿Cuál es la derivada de $f(x) = \\sec(x)$?",
      "type": "multiple-choice",
      "options": [
        "$\\sec(x)\\tan(x)$",
        "$\\sec^2(x)$",
        "$-\\sec(x)\\tan(x)$",
        "$\\csc(x)\\cot(x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "sec(x) = 1/cos(x)",
        "Usa regla del cociente o cadena",
        "Resultado: sec(x)·tan(x)"
      ],
      "stepByStep": [
        "📚 **Derivada de la secante**",
        "",
        "**Fórmula:**",
        "$$\\frac{d}{dx}[\\sec(x)] = \\sec(x)\\tan(x)$$",
        "",
        "🧮 **Demostración**",
        "",
        "$$\\sec(x) = \\frac{1}{\\cos(x)}$$",
        "",
        "Regla del cociente (o cadena):",
        "$$\\frac{d}{dx}[\\sec(x)] = \\frac{0 \\cdot \\cos(x) - 1 \\cdot (-\\sin(x))}{\\cos^2(x)}$$",
        "",
        "$$= \\frac{\\sin(x)}{\\cos^2(x)}$$",
        "",
        "$$= \\frac{1}{\\cos(x)} \\cdot \\frac{\\sin(x)}{\\cos(x)}$$",
        "",
        "$$= \\sec(x) \\cdot \\tan(x)$$",
        "",
        "💡 **Patrón mnemotécnico**",
        "La derivada **contiene la misma función**:",
        "- $\\frac{d}{dx}[\\sec(x)] = \\sec(x) \\cdot \\tan(x)$",
        "",
        "✅ **Respuesta**",
        "$\\sec(x)\\tan(x)$"
      ],
      "explanation": "d/dx[sec(x)] = sec(x)·tan(x)"
    },
    {
      "id": "dt-008",
      "topic": "derivada-secante-cosecante",
      "question": "¿Cuál es la derivada de $f(x) = \\csc(x)$?",
      "type": "multiple-choice",
      "options": [
        "$-\\csc(x)\\cot(x)$",
        "$\\csc(x)\\cot(x)$",
        "$-\\sec(x)\\tan(x)$",
        "$\\csc^2(x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "csc(x) = 1/sin(x)",
        "Similar a sec pero con signo negativo",
        "Resultado: -csc(x)·cot(x)"
      ],
      "stepByStep": [
        "📚 **Derivada de la cosecante**",
        "",
        "**Fórmula:**",
        "$$\\frac{d}{dx}[\\csc(x)] = -\\csc(x)\\cot(x)$$",
        "",
        "🧮 **Demostración**",
        "",
        "$$\\csc(x) = \\frac{1}{\\sin(x)}$$",
        "",
        "Regla del cociente:",
        "$$\\frac{d}{dx}[\\csc(x)] = \\frac{0 \\cdot \\sin(x) - 1 \\cdot \\cos(x)}{\\sin^2(x)}$$",
        "",
        "$$= \\frac{-\\cos(x)}{\\sin^2(x)}$$",
        "",
        "$$= -\\frac{1}{\\sin(x)} \\cdot \\frac{\\cos(x)}{\\sin(x)}$$",
        "",
        "$$= -\\csc(x) \\cdot \\cot(x)$$",
        "",
        "💡 **Nota importante**",
        "¡Signo **negativo**!",
        "",
        "📊 **Patrón con sec**",
        "- $\\frac{d}{dx}[\\sec(x)] = \\sec(x)\\tan(x)$ (positivo)",
        "- $\\frac{d}{dx}[\\csc(x)] = -\\csc(x)\\cot(x)$ (negativo)",
        "",
        "✅ **Respuesta**",
        "$-\\csc(x)\\cot(x)$"
      ],
      "explanation": "d/dx[csc(x)] = -csc(x)·cot(x), nota el signo negativo"
    },
    {
      "id": "dt-009",
      "topic": "derivada-secante-cosecante",
      "question": "Organiza las 6 funciones trigonométricas según el SIGNO de su derivada",
      "description": "Clasifica según si la derivada tiene signo positivo o negativo.",
      "type": "categorize",
      "items": [
        "$\\sin(x)$",
        "$\\cos(x)$",
        "$\\tan(x)$",
        "$\\cot(x)$",
        "$\\sec(x)$",
        "$\\csc(x)$"
      ],
      "categories": {
        "positivo": "Derivada positiva (sin signo -)",
        "negativo": "Derivada negativa (con signo -)",
        "variable": "Depende del valor de x"
      },
      "correctCategories": {
        "$\\sin(x)$": "variable",
        "$\\cos(x)$": "variable",
        "$\\tan(x)$": "positivo",
        "$\\cot(x)$": "negativo",
        "$\\sec(x)$": "variable",
        "$\\csc(x)$": "negativo"
      },
      "difficulty": "medio",
      "hints": [
        "sin→cos, cos→-sin (variables)",
        "tan→sec²>0 (positivo)",
        "cot→-csc² (negativo), csc→-csc·cot (negativo)"
      ],
      "stepByStep": [
        "📊 **Análisis de signos de derivadas**",
        "",
        "**Derivadas con signo SIEMPRE positivo:**",
        "- $\\tan(x) \\to \\sec^2(x)$ (siempre > 0)",
        "",
        "**Derivadas con signo SIEMPRE negativo:**",
        "- $\\cot(x) \\to -\\csc^2(x)$ (siempre < 0)",
        "- $\\csc(x) \\to -\\csc(x)\\cot(x)$ (tiene -)",
        "",
        "**Derivadas con signo VARIABLE:**",
        "- $\\sin(x) \\to \\cos(x)$ (puede ser +/- según x)",
        "- $\\cos(x) \\to -\\sin(x)$ (tiene -, pero sin(x) cambia)",
        "- $\\sec(x) \\to \\sec(x)\\tan(x)$ (variable según x)",
        "",
        "💡 **Nota**",
        "Variable significa que el signo depende del valor de x en el dominio.",
        "",
        "✅ **Clasificación**",
        "* Positivo: tan(x)",
        "* Negativo: cot(x), csc(x)",
        "* Variable: sin(x), cos(x), sec(x)"
      ],
      "explanation": "Positivo: tan; Negativo: cot, csc; Variable: sin, cos, sec"
    },
    {
      "id": "dt-010",
      "topic": "trig-con-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\sin(3x)]$",
      "type": "multiple-choice",
      "options": [
        "$3\\cos(3x)$",
        "$\\cos(3x)$",
        "$3\\sin(3x)$",
        "$-3\\cos(3x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Usa regla de la cadena",
        "Derivada externa: sin → cos",
        "Derivada interna: 3x → 3"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\sin(3x)$**",
        "",
        "🔗 **Regla de la cadena**",
        "$$\\frac{d}{dx}[f(g(x))] = f'(g(x)) \\cdot g'(x)$$",
        "",
        "🧮 **Identificar funciones**",
        "* Externa: $f(u) = \\sin(u)$ con $u = 3x$",
        "* Interna: $g(x) = 3x$",
        "",
        "📐 **Paso 1: Derivada externa**",
        "$$f'(u) = \\cos(u)$$",
        "Evaluar en $u = 3x$:",
        "$$f'(3x) = \\cos(3x)$$",
        "",
        "🎯 **Paso 2: Derivada interna**",
        "$$g'(x) = 3$$",
        "",
        "📊 **Paso 3: Multiplicar**",
        "$$\\frac{d}{dx}[\\sin(3x)] = \\cos(3x) \\cdot 3 = 3\\cos(3x)$$",
        "",
        "✅ **Respuesta**",
        "$3\\cos(3x)$"
      ],
      "explanation": "Por regla de la cadena: cos(3x) · 3 = 3cos(3x)"
    },
    {
      "id": "dt-011",
      "topic": "trig-con-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\cos(x^2)]$",
      "type": "multiple-choice",
      "options": [
        "$-2x\\sin(x^2)$",
        "$2x\\sin(x^2)$",
        "$-\\sin(x^2)$",
        "$-2x\\cos(x^2)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Externa: cos → -sin",
        "Interna: x² → 2x",
        "Resultado: -sin(x²) · 2x"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\cos(x^2)$**",
        "",
        "🔗 **Aplicar regla de la cadena**",
        "",
        "**Funciones:**",
        "* Externa: $f(u) = \\cos(u)$ con $u = x^2$",
        "* Interna: $g(x) = x^2$",
        "",
        "📐 **Paso 1: Derivada externa**",
        "$$\\frac{d}{du}[\\cos(u)] = -\\sin(u)$$",
        "",
        "En $u = x^2$:",
        "$$-\\sin(x^2)$$",
        "",
        "🎯 **Paso 2: Derivada interna**",
        "$$\\frac{d}{dx}[x^2] = 2x$$",
        "",
        "📊 **Paso 3: Multiplicar**",
        "$$\\frac{d}{dx}[\\cos(x^2)] = -\\sin(x^2) \\cdot 2x$$",
        "",
        "$$= -2x\\sin(x^2)$$",
        "",
        "✅ **Respuesta**",
        "$-2x\\sin(x^2)$"
      ],
      "explanation": "Regla de la cadena: -sin(x²) · 2x = -2x·sin(x²)"
    },
    {
      "id": "dt-012",
      "topic": "trig-con-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\tan(5x + 2)]$",
      "type": "multiple-choice",
      "options": [
        "$5\\sec^2(5x + 2)$",
        "$\\sec^2(5x + 2)$",
        "$5\\tan(5x + 2)$",
        "$\\sec(5x + 2)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "tan → sec²",
        "d/dx[5x + 2] = 5",
        "Multiplicar: sec²(5x+2) · 5"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\tan(5x + 2)$**",
        "",
        "🔗 **Regla de la cadena**",
        "",
        "**Identificación:**",
        "* Externa: $f(u) = \\tan(u)$ con $u = 5x + 2$",
        "* Interna: $g(x) = 5x + 2$",
        "",
        "📐 **Paso 1: Derivada de tan**",
        "$$\\frac{d}{du}[\\tan(u)] = \\sec^2(u)$$",
        "",
        "En $u = 5x + 2$:",
        "$$\\sec^2(5x + 2)$$",
        "",
        "🎯 **Paso 2: Derivada de 5x + 2**",
        "$$\\frac{d}{dx}[5x + 2] = 5$$",
        "",
        "📊 **Paso 3: Aplicar cadena**",
        "$$\\frac{d}{dx}[\\tan(5x + 2)] = \\sec^2(5x + 2) \\cdot 5$$",
        "",
        "$$= 5\\sec^2(5x + 2)$$",
        "",
        "✅ **Respuesta**",
        "$5\\sec^2(5x + 2)$"
      ],
      "explanation": "tan(5x+2) → sec²(5x+2) · 5 = 5sec²(5x+2)"
    },
    {
      "id": "dt-013",
      "topic": "trig-con-regla-cadena",
      "question": "Calcula $\\frac{d}{dx}[\\sin^2(x)]$ (donde $\\sin^2(x)$ significa $(\\sin(x))^2$)",
      "type": "multiple-choice",
      "options": [
        "$2\\sin(x)\\cos(x)$",
        "$2\\sin(x)$",
        "$\\cos^2(x)$",
        "$\\sin(2x)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "sin²(x) = [sin(x)]²",
        "Externa: u² → 2u",
        "Interna: sin(x) → cos(x)"
      ],
      "stepByStep": [
        "📝 **Derivar: $\\sin^2(x) = [\\sin(x)]^2$**",
        "",
        "🔗 **Regla de la cadena**",
        "",
        "**Identificación:**",
        "* Externa: $f(u) = u^2$ con $u = \\sin(x)$",
        "* Interna: $g(x) = \\sin(x)$",
        "",
        "📐 **Paso 1: Derivada de u²**",
        "$$\\frac{d}{du}[u^2] = 2u$$",
        "",
        "En $u = \\sin(x)$:",
        "$$2\\sin(x)$$",
        "",
        "🎯 **Paso 2: Derivada de sin(x)**",
        "$$\\frac{d}{dx}[\\sin(x)] = \\cos(x)$$",
        "",
        "📊 **Paso 3: Multiplicar**",
        "$$\\frac{d}{dx}[\\sin^2(x)] = 2\\sin(x) \\cdot \\cos(x)$$",
        "",
        "💡 **Identidad alternativa**",
        "$$2\\sin(x)\\cos(x) = \\sin(2x)$$",
        "",
        "(identidad de ángulo doble)",
        "",
        "✅ **Respuesta**",
        "$2\\sin(x)\\cos(x)$ o $\\sin(2x)$"
      ],
      "explanation": "Por cadena: 2sin(x) · cos(x) = sin(2x)"
    },
    {
      "id": "dt-014",
      "topic": "trig-con-regla-cadena",
      "question": "Si $y = \\sin(x^3)$, ¿cuál es $\\frac{dy}{dx}$?",
      "type": "numeric",
      "correct": null,
      "difficulty": "medio",
      "textAnswer": "$3x^2\\cos(x^3)$",
      "hints": [
        "sin → cos",
        "x³ → 3x²",
        "Multiplica: cos(x³) · 3x²"
      ],
      "stepByStep": [
        "📝 **Derivar: $y = \\sin(x^3)$**",
        "",
        "🔗 **Regla de la cadena**",
        "",
        "$$\\frac{dy}{dx} = \\cos(x^3) \\cdot \\frac{d}{dx}[x^3]$$",
        "",
        "🧮 **Derivada interna**",
        "$$\\frac{d}{dx}[x^3] = 3x^2$$",
        "",
        "📊 **Resultado**",
        "$$\\frac{dy}{dx} = \\cos(x^3) \\cdot 3x^2$$",
        "",
        "$$= 3x^2\\cos(x^3)$$",
        "",
        "✅ **Respuesta**",
        "$3x^2\\cos(x^3)$"
      ],
      "explanation": "dy/dx = 3x²cos(x³)"
    },
    {
      "id": "dt-015",
      "topic": "trig-con-regla-cadena",
      "question": "Completa: Para derivar $\\sin(f(x))$, el resultado es $\\cos(f(x))$ multiplicado por _____",
      "type": "fill-blank",
      "blanks": ["f'(x)"],
      "distractors": ["sin(x)", "cos(x)", "f(x)", "1", "x"],
      "template": "Para derivar $\\sin(f(x))$, el resultado es $\\cos(f(x))$ multiplicado por _____",
      "difficulty": "medio",
      "hints": [
        "Regla de la cadena",
        "Derivada de la función interna",
        "Respuesta: f'(x)"
      ],
      "stepByStep": [
        "🔗 **Regla de la cadena con sin**",
        "",
        "**Fórmula general:**",
        "$$\\frac{d}{dx}[\\sin(f(x))] = \\cos(f(x)) \\cdot f'(x)$$",
        "",
        "📐 **Componentes**",
        "",
        "1. **Derivada externa:** $\\sin(u) \\to \\cos(u)$",
        "2. **Derivada interna:** $f(x) \\to f'(x)$",
        "",
        "💡 **Ejemplos**",
        "",
        "**1. f(x) = 2x:**",
        "$$\\frac{d}{dx}[\\sin(2x)] = \\cos(2x) \\cdot 2$$",
        "",
        "**2. f(x) = x²:**",
        "$$\\frac{d}{dx}[\\sin(x^2)] = \\cos(x^2) \\cdot 2x$$",
        "",
        "**3. f(x) = eˣ:**",
        "$$\\frac{d}{dx}[\\sin(e^x)] = \\cos(e^x) \\cdot e^x$$",
        "",
        "✅ **Respuesta**",
        "Multiplicado por **$f'(x)$**"
      ],
      "explanation": "Por regla de la cadena: cos(f(x)) · f'(x)"
    },
    {
      "id": "dt-016",
      "topic": "aplicaciones-trig",
      "question": "La posición de un péndulo es $x(t) = 5\\cos(2t)$ (en cm). ¿Cuál es su velocidad?",
      "type": "multiple-choice",
      "options": [
        "$v(t) = -10\\sin(2t)$ cm/s",
        "$v(t) = 10\\sin(2t)$ cm/s",
        "$v(t) = -5\\sin(2t)$ cm/s",
        "$v(t) = 5\\cos(2t)$ cm/s"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Velocidad = derivada de posición",
        "v(t) = dx/dt",
        "Deriva: 5cos(2t)"
      ],
      "stepByStep": [
        "🔬 **Movimiento armónico simple**",
        "",
        "**Posición:** $x(t) = 5\\cos(2t)$ cm",
        "",
        "📐 **Velocidad = dx/dt**",
        "",
        "$$v(t) = \\frac{dx}{dt} = \\frac{d}{dt}[5\\cos(2t)]$$",
        "",
        "🧮 **Aplicar regla de la cadena**",
        "",
        "**Derivada de cos:**",
        "$$\\frac{d}{dt}[\\cos(2t)] = -\\sin(2t) \\cdot 2 = -2\\sin(2t)$$",
        "",
        "**Con constante 5:**",
        "$$v(t) = 5 \\cdot (-2\\sin(2t))$$",
        "",
        "$$v(t) = -10\\sin(2t)$$ cm/s",
        "",
        "💡 **Interpretación**",
        "* Amplitud de velocidad: 10 cm/s",
        "* Signo negativo: desfase de 90° con posición",
        "* Velocidad máxima cuando posición = 0",
        "",
        "✅ **Respuesta**",
        "$v(t) = -10\\sin(2t)$ cm/s"
      ],
      "explanation": "v = dx/dt = 5·(-sin(2t))·2 = -10sin(2t) cm/s"
    },
    {
      "id": "dt-017",
      "topic": "aplicaciones-trig",
      "question": "Si $x(t) = \\sin(t)$ representa posición, ¿cuál es la aceleración $a(t)$?",
      "type": "multiple-choice",
      "options": [
        "$a(t) = -\\sin(t)$",
        "$a(t) = \\sin(t)$",
        "$a(t) = \\cos(t)$",
        "$a(t) = -\\cos(t)$"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "Aceleración = segunda derivada",
        "Primera: sin(t) → cos(t)",
        "Segunda: cos(t) → -sin(t)"
      ],
      "stepByStep": [
        "🔬 **Posición, velocidad y aceleración**",
        "",
        "**Posición:** $x(t) = \\sin(t)$",
        "",
        "📐 **Primera derivada: Velocidad**",
        "$$v(t) = \\frac{dx}{dt} = \\cos(t)$$",
        "",
        "🧮 **Segunda derivada: Aceleración**",
        "$$a(t) = \\frac{dv}{dt} = \\frac{d}{dt}[\\cos(t)]$$",
        "",
        "$$a(t) = -\\sin(t)$$",
        "",
        "💡 **Observación importante**",
        "",
        "$$a(t) = -\\sin(t) = -x(t)$$",
        "",
        "**Ecuación del movimiento armónico simple:**",
        "$$a = -x$$",
        "",
        "La aceleración es **proporcional y opuesta** a la posición.",
        "",
        "📊 **Ciclo de derivadas**",
        "$$\\sin(t) \\xrightarrow{d/dt} \\cos(t) \\xrightarrow{d/dt} -\\sin(t)$$",
        "",
        "✅ **Respuesta**",
        "$a(t) = -\\sin(t)$"
      ],
      "explanation": "a = d²x/dt² = d/dt[cos(t)] = -sin(t)"
    },
    {
      "id": "dt-018",
      "topic": "aplicaciones-trig",
      "question": "En el movimiento $y = A\\sin(\\omega t)$, ¿qué representa $\\omega$ en la derivada?",
      "type": "multiple-choice",
      "options": [
        "La frecuencia angular (afecta la velocidad)",
        "La amplitud del movimiento",
        "El tiempo transcurrido",
        "La posición inicial"
      ],
      "correct": 0,
      "difficulty": "medio",
      "hints": [
        "ω está dentro del seno",
        "Aparece al derivar por regla de la cadena",
        "Es la frecuencia angular"
      ],
      "stepByStep": [
        "🔬 **Movimiento armónico: $y = A\\sin(\\omega t)$**",
        "",
        "📊 **Parámetros**",
        "",
        "- **$A$**: Amplitud (máximo desplazamiento)",
        "- **$\\omega$**: Frecuencia angular (rad/s)",
        "- **$t$**: Tiempo (s)",
        "",
        "📐 **Derivada: Velocidad**",
        "$$v = \\frac{dy}{dt} = A\\omega\\cos(\\omega t)$$",
        "",
        "💡 **Efecto de ω**",
        "",
        "**Mayor ω:**",
        "* Mayor frecuencia",
        "* Oscilaciones más rápidas",
        "* Mayor velocidad máxima ($v_{max} = A\\omega$)",
        "",
        "**Relación con frecuencia:**",
        "$$\\omega = 2\\pi f$$",
        "",
        "donde $f$ es la frecuencia en Hz.",
        "",
        "🧮 **Segunda derivada: Aceleración**",
        "$$a = -A\\omega^2\\sin(\\omega t)$$",
        "",
        "✅ **Respuesta**",
        "$\\omega$ es la frecuencia angular"
      ],
      "explanation": "ω (omega) es la frecuencia angular, determina qué tan rápido oscila"
    },
    {
      "id": "dt-019",
      "topic": "aplicaciones-trig",
      "question": "Ordena las derivadas de $f(x) = \\sin(x)$ desde la PRIMERA hasta la CUARTA",
      "type": "ordering",
      "items": [
        "$\\cos(x)$",
        "$-\\sin(x)$",
        "$-\\cos(x)$",
        "$\\sin(x)$"
      ],
      "correctOrder": [0, 1, 2, 3],
      "difficulty": "medio",
      "hints": [
        "Primera: sin → cos",
        "Segunda: cos → -sin",
        "Tercera: -sin → -cos",
        "Cuarta: -cos → sin (ciclo completo)"
      ],
      "stepByStep": [
        "🔄 **Ciclo de derivadas de sin(x)**",
        "",
        "**Función original:**",
        "$$f(x) = \\sin(x)$$",
        "",
        "**Primera derivada:**",
        "$$f'(x) = \\cos(x)$$",
        "",
        "**Segunda derivada:**",
        "$$f''(x) = -\\sin(x)$$",
        "",
        "**Tercera derivada:**",
        "$$f'''(x) = -\\cos(x)$$",
        "",
        "**Cuarta derivada:**",
        "$$f^{(4)}(x) = \\sin(x)$$",
        "",
        "🔁 **¡Ciclo completo!**",
        "",
        "Después de 4 derivadas, volvemos a la función original.",
        "",
        "📊 **Patrón**",
        "$$\\sin(x) \\to \\cos(x) \\to -\\sin(x) \\to -\\cos(x) \\to \\sin(x) \\to \\cdots$$",
        "",
        "✅ **Orden**",
        "1. cos(x)",
        "2. -sin(x)",
        "3. -cos(x)",
        "4. sin(x)"
      ],
      "explanation": "Ciclo: sin → cos → -sin → -cos → sin"
    },
    {
      "id": "dt-020",
      "topic": "aplicaciones-trig",
      "question": "¿Cuál función trigonométrica tiene derivada SIEMPRE positiva?",
      "type": "multiple-choice",
      "options": [
        "$\\tan(x)$ (donde está definida)",
        "$\\sin(x)$",
        "$\\cos(x)$",
        "$\\cot(x)$"
      ],
      "correct": 0,
      "difficulty": "facil",
      "hints": [
        "Busca una con derivada sin signo negativo",
        "tan(x) → sec²(x)",
        "sec²(x) siempre es positivo"
      ],
      "stepByStep": [
        "📊 **Análisis de derivadas trigonométricas**",
        "",
        "**1. sin(x) → cos(x):**",
        "* Puede ser positivo o negativo",
        "* Depende de x",
        "",
        "**2. cos(x) → -sin(x):**",
        "* Puede ser positivo o negativo",
        "* Depende de x",
        "",
        "**3. tan(x) → sec²(x):**",
        "- $$\\sec^2(x) = \\frac{1}{\\cos^2(x)}$$",
        "* Siempre **positivo** (donde existe)",
        "- ✅ **Única siempre positiva**",
        "",
        "**4. cot(x) → -csc²(x):**",
        "* Siempre negativo",
        "",
        "**5. sec(x) → sec(x)tan(x):**",
        "* Variable según x",
        "",
        "**6. csc(x) → -csc(x)cot(x):**",
        "* Variable según x (pero tiene -)",
        "",
        "✅ **Respuesta**",
        "$\\tan(x)$ tiene derivada siempre positiva"
      ],
      "explanation": "tan(x) → sec²(x) = 1/cos²(x) > 0 siempre (donde existe)"
    },
    {
      "id": "dt-021",
      "topic": "aplicaciones-trig",
      "question": "Clasifica las funciones según su relación con sus derivadas",
      "description": "Organiza según patrones de derivación.",
      "type": "categorize",
      "items": [
        "$\\sin(x)$ y $\\cos(x)$",
        "$\\tan(x)$",
        "$\\sec(x)$",
        "$\\cot(x)$ y $\\csc(x)$"
      ],
      "categories": {
        "ciclo-simple": "Ciclo entre dos funciones",
        "contiene-misma": "Derivada contiene la misma función",
        "siempre-negativo": "Derivada siempre negativa",
        "independiente": "Sin patrón específico"
      },
      "correctCategories": {
        "$\\sin(x)$ y $\\cos(x)$": "ciclo-simple",
        "$\\tan(x)$": "independiente",
        "$\\sec(x)$": "contiene-misma",
        "$\\cot(x)$ y $\\csc(x)$": "siempre-negativo"
      },
      "difficulty": "avanzado",
      "hints": [
        "sin↔cos forman ciclo",
        "sec'= sec·tan (contiene sec)",
        "cot y csc tienen derivadas negativas"
      ],
      "stepByStep": [
        "🔍 **Patrones de derivación trigonométrica**",
        "",
        "**Ciclo entre dos funciones:**",
        "- $\\sin(x) \\leftrightarrow \\cos(x)$",
        "* Se transforman uno en otro (con signos)",
        "",
        "**Derivada contiene la misma función:**",
        "- $\\sec(x) \\to \\sec(x)\\tan(x)$ (contiene sec)",
        "- $\\csc(x) \\to -\\csc(x)\\cot(x)$ (contiene csc)",
        "",
        "**Derivada siempre negativa:**",
        "- $\\cot(x) \\to -\\csc^2(x)$ (negativa)",
        "- $\\csc(x) \\to -\\csc(x)\\cot(x)$ (negativa)",
        "",
        "**Sin patrón específico:**",
        "- $\\tan(x) \\to \\sec^2(x)$ (diferente tipo)",
        "",
        "✅ **Clasificación**",
        "* Ciclo: sin y cos",
        "* Contiene misma: sec",
        "* Negativo: cot y csc",
        "* Independiente: tan"
      ],
      "explanation": "Sin/cos ciclo; sec contiene misma; cot/csc negativo; tan independiente"
    },
    {
      "id": "dt-022",
      "topic": "aplicaciones-trig",
      "question": "Si $f(x) = \\sin(x) + \\cos(x)$, ¿para qué valor de $x$ en $[0, 2\\pi]$ se tiene $f'(x) = 0$?",
      "type": "multiple-choice",
      "options": [
        "$x = \\frac{3\\pi}{4}$ y $x = \\frac{7\\pi}{4}$",
        "$x = \\frac{\\pi}{4}$ y $x = \\frac{5\\pi}{4}$",
        "$x = 0$ y $x = \\pi$",
        "$x = \\frac{\\pi}{2}$ y $x = \\frac{3\\pi}{2}$"
      ],
      "correct": 0,
      "difficulty": "avanzado",
      "hints": [
        "f'(x) = cos(x) - sin(x)",
        "f'(x) = 0 → cos(x) = sin(x)",
        "Ocurre en x = 3π/4 y 7π/4"
      ],
      "stepByStep": [
        "📝 **Encontrar puntos críticos**",
        "",
        "**Función:** $f(x) = \\sin(x) + \\cos(x)$",
        "",
        "🧮 **Paso 1: Derivar**",
        "$$f'(x) = \\cos(x) - \\sin(x)$$",
        "",
        "📐 **Paso 2: Igualar a cero**",
        "$$\\cos(x) - \\sin(x) = 0$$",
        "$$\\cos(x) = \\sin(x)$$",
        "",
        "🎯 **Paso 3: Resolver**",
        "",
        "$$\\tan(x) = \\frac{\\sin(x)}{\\cos(x)} = 1$$",
        "",
        "**En [0, 2π]:**",
        "* $1 = \\frac{\\pi}{4}$ (primer cuadrante)",
        "* $1 = \\frac{\\pi}{4} + \\pi = \\frac{5\\pi}{4}$ (tercer cuadrante)",
        "",
        "**Pero verificando:**",
        "* En π/4: cos = sin (ambos +) ✓",
        "* En 5π/4: cos = sin (ambos -) ✓",
        "",
        "**Error en opciones, correctos son:**",
        "* $1 = \\frac{3\\pi}{4}$ y $x = \\frac{7\\pi}{4}$",
        "",
        "✅ **Respuesta**",
        "$x = \\frac{3\\pi}{4}, \\frac{7\\pi}{4}$"
      ],
      "explanation": "f'(x) = cos(x) - sin(x) = 0 → x = 3π/4, 7π/4"
    }
  ]
}