{ "query": { "display": "derivada de $$x\\sqrt{4-x^{2}}$$", "symbolab_question": "PRE_CALC#derivative x\\sqrt{4-x^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{-2x^{2}+4}{\\sqrt{4-x^{2}}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x\\sqrt{4-x^{2}}\\right)=\\frac{-2x^{2}+4}{\\sqrt{4-x^{2}}}$$", "input": "\\frac{d}{dx}\\left(x\\sqrt{4-x^{2}}\\right)", "steps": [ { "type": "step", "primary": "Aplicar la regla del producto: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x,\\:g=\\sqrt{4-x^{2}}$$" ], "result": "=\\frac{dx}{dx}\\sqrt{4-x^{2}}+\\frac{d}{dx}\\left(\\sqrt{4-x^{2}}\\right)x", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Aplicar la regla de derivación: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{4-x^{2}}\\right)=-\\frac{x}{\\sqrt{4-x^{2}}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{4-x^{2}}\\right)", "steps": [ { "type": "interim", "title": "Aplicar la regla de la cadena:$${\\quad}\\frac{1}{2\\sqrt{4-x^{2}}}\\frac{d}{dx}\\left(4-x^{2}\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{4-x^{2}}\\right)", "result": "=\\frac{1}{2\\sqrt{4-x^{2}}}\\frac{d}{dx}\\left(4-x^{2}\\right)", "steps": [ { "type": "step", "primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=4-x^{2}$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(4-x^{2}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\sqrt{u}\\right)=\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{d}{du}\\left(\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Aplicar las leyes de los exponentes: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{du}\\left(u^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Aplicar la regla de la potencia: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{2}u^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplificar $$\\frac{1}{2}u^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{1}{2}u^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{u}}", "steps": [ { "type": "interim", "title": "$$u^{\\frac{1}{2}-1}=u^{-\\frac{1}{2}}$$", "input": "u^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Simplificar $$\\frac{1}{2}-1\\:$$en una fracción:$${\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=u^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convertir a fracción: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{2}" }, { "type": "step", "primary": "Ya que los denominadores son iguales, combinar las fracciones: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiplicar los numeros: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Sumar/restar lo siguiente: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Aplicar las propiedades de las fracciones: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKe0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjMvOxDqXzE3/CFO0TFmffHAH2kDe5DGYTz3TrPquGdIhyukSOA/1RgMKO0TMhInPOMabdUggEogUL9RT7PNKh0VQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}u^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Aplicar las leyes de los exponentes: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{u}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{u}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiplicar fracciones: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{u}}" }, { "type": "step", "primary": "Multiplicar los numeros: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{u}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JOPQ2g2GS9EQptV8nckZSrH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJv2RkT96g5Q5jVbn5fyeQzwB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC0fckB6tn+Lslm20bk1NfC+RFXBEVoMC309dBjB0EbJM4Es6agjDQJIYZAr5O37aAQ==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{u}}\\frac{d}{dx}\\left(4-x^{2}\\right)" }, { "type": "step", "primary": "Sustituir en la ecuación $$u=4-x^{2}$$", "result": "=\\frac{1}{2\\sqrt{4-x^{2}}}\\frac{d}{dx}\\left(4-x^{2}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqws7SQcG//90aDq2y0nVyst9MMddrkFuNTLaPWMeYe3Z3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJXv++bSprT8DRLjDQza+XRVK8bX5dVPv1mw7bndgYyo4aKwxODUNSw+sWQrjsE7uIE5Nrp7QlQvtOoqamnUHSv5RSpN33oxZMojoqvYhvSJANJ77zFVgozt2myZ932vHaY8M3qPbRt2Bhfd2B52Djj+79Tf85JKR+wURjZIcdFtqg==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4-x^{2}\\right)=-2x$$", "input": "\\frac{d}{dx}\\left(4-x^{2}\\right)", "steps": [ { "type": "step", "primary": "Aplicar la regla de la suma/diferencia: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(4\\right)-\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\right)", "steps": [ { "type": "step", "primary": "Derivada de una constante: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Aplicar la regla de la potencia: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplificar", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "step", "result": "=0-2x" }, { "type": "step", "primary": "Simplificar", "result": "=-2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{4-x^{2}}}\\left(-2x\\right)" }, { "type": "interim", "title": "Simplificar $$\\frac{1}{2\\sqrt{4-x^{2}}}\\left(-2x\\right):{\\quad}-\\frac{x}{\\sqrt{4-x^{2}}}$$", "input": "\\frac{1}{2\\sqrt{4-x^{2}}}\\left(-2x\\right)", "result": "=-\\frac{x}{\\sqrt{4-x^{2}}}", "steps": [ { "type": "step", "primary": "Quitar los parentesis: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2\\sqrt{4-x^{2}}}\\cdot\\:2x" }, { "type": "step", "primary": "Multiplicar fracciones: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:2x}{2\\sqrt{4-x^{2}}}" }, { "type": "step", "primary": "Eliminar los terminos comunes: $$2$$", "result": "=-\\frac{1\\cdot\\:x}{\\sqrt{4-x^{2}}}" }, { "type": "step", "primary": "Multiplicar: $$1\\cdot\\:x=x$$", "result": "=-\\frac{x}{\\sqrt{-x^{2}+4}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13C05eUZ+miyV2wo6fK1k/Zk6O6Dd3zpuSnKXNDXc0BYg4ebTKSUwQXHFFpEsrzHtUqUlp37JZxv9+CIGyLDYI+uDk5PyMmrrndFr6EuaTQELKgQUxJPyUNnGfVirkcwpVO4DWU6bOJf1kDNjB36N9KhyY2KuYTFWgaIvKFCH8XXOk9dOCBrKP/hY2VXiQndH6n+IASZeFjDtawNGt9P21GJY=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=1\\cdot\\:\\sqrt{4-x^{2}}+\\left(-\\frac{x}{\\sqrt{4-x^{2}}}\\right)x" }, { "type": "interim", "title": "Simplificar $$1\\cdot\\:\\sqrt{4-x^{2}}+\\left(-\\frac{x}{\\sqrt{4-x^{2}}}\\right)x:{\\quad}\\frac{-2x^{2}+4}{\\sqrt{4-x^{2}}}$$", "input": "1\\cdot\\:\\sqrt{4-x^{2}}+\\left(-\\frac{x}{\\sqrt{4-x^{2}}}\\right)x", "result": "=\\frac{-2x^{2}+4}{\\sqrt{4-x^{2}}}", "steps": [ { "type": "step", "primary": "Quitar los parentesis: $$\\left(-a\\right)=-a$$", "result": "=1\\cdot\\:\\sqrt{4-x^{2}}-\\frac{x}{\\sqrt{4-x^{2}}}x" }, { "type": "interim", "title": "$$1\\cdot\\:\\sqrt{4-x^{2}}=\\sqrt{4-x^{2}}$$", "input": "1\\cdot\\:\\sqrt{4-x^{2}}", "steps": [ { "type": "step", "primary": "Multiplicar: $$1\\cdot\\:\\sqrt{4-x^{2}}=\\sqrt{4-x^{2}}$$", "result": "=\\sqrt{4-x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UHFhuJ6yJNDOcGH7Fz25/+bQ8eq9KPndPsh+cpcAozgDnzlbPZjyKgy1eUCFsLd5FoA/NZPhsKr7yKCV92V7VKBtZyu+LaYvasa3tFjSwOE5S/rzXhIImTcNKflhJ16Ki9jSJ04Wg+AegNbFs57a0rcwG4qVJJt4DwPN4pwAaPo=" } }, { "type": "interim", "title": "$$\\frac{x}{\\sqrt{4-x^{2}}}x=\\frac{x^{2}}{\\sqrt{4-x^{2}}}$$", "input": "\\frac{x}{\\sqrt{4-x^{2}}}x", "steps": [ { "type": "step", "primary": "Multiplicar fracciones: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{xx}{\\sqrt{4-x^{2}}}" }, { "type": "interim", "title": "$$xx=x^{2}$$", "input": "xx", "steps": [ { "type": "step", "primary": "Aplicar las leyes de los exponentes: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Sumar: $$1+1=2$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74tglIeOAQqOR+OcPiyzQUczBWJotReR4P4m6RE6FZ2M7Aq6fHyeqJtW5OKbXVcT+IBF/biSmVq3Z2pV/8nBrAiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\frac{x^{2}}{\\sqrt{4-x^{2}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73Y7dsHR5Q/6nneLHJZRjGZ2OT8eCwMwHqi5FHuTWe8vehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgprPMEHM/mmc4OdEqEQA2YbJTENVM5lrkpLfucRPZru95mJW+vqX35fV9zI/bFAf9gAKWVz0ENdilMvqJ8y7Vc/bnHaVVCnUhcvup9B6HTXulApZXPQQ12KUy+onzLtVz9hZFQ8OchBdgxMOJrEMyk3s=" } }, { "type": "step", "result": "=\\sqrt{-x^{2}+4}-\\frac{x^{2}}{\\sqrt{-x^{2}+4}}" }, { "type": "step", "primary": "Convertir a fracción: $$\\sqrt{-x^{2}+4}=\\frac{\\sqrt{4-x^{2}}\\sqrt{4-x^{2}}}{\\sqrt{4-x^{2}}}$$", "result": "=-\\frac{x^{2}}{\\sqrt{4-x^{2}}}+\\frac{\\sqrt{4-x^{2}}\\sqrt{4-x^{2}}}{\\sqrt{4-x^{2}}}" }, { "type": "step", "primary": "Ya que los denominadores son iguales, combinar las fracciones: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-x^{2}+\\sqrt{4-x^{2}}\\sqrt{4-x^{2}}}{\\sqrt{4-x^{2}}}" }, { "type": "interim", "title": "$$-x^{2}+\\sqrt{4-x^{2}}\\sqrt{4-x^{2}}=-2x^{2}+4$$", "input": "-x^{2}+\\sqrt{4-x^{2}}\\sqrt{4-x^{2}}", "steps": [ { "type": "step", "primary": "Aplicar las leyes de los exponentes: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{-x^{2}+4}\\sqrt{-x^{2}+4}=4-x^{2}$$" ], "result": "=-x^{2}+4-x^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Agrupar términos semejantes", "result": "=-x^{2}-x^{2}+4" }, { "type": "step", "primary": "Sumar elementos similares: $$-x^{2}-x^{2}=-2x^{2}$$", "result": "=-2x^{2}+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jaxItqg/p85u4UD5o36ZlCgh+5qTaXZJcH7xpUJp/Uob0QnaQXeEGehdJB+LYyEKCUCWbkwGOY7PqKo3U/JLJaSePBlrMSalya/cb6+i9nV8PILidLzxItlzEwp/VwP73VaLVwqT5GHTOa2VgvEsA1hrrK9mUbBIbL7c4y4Kk5heZuk4A0LKlR1fHxRTdzqY" } }, { "type": "step", "result": "=\\frac{-2x^{2}+4}{\\sqrt{4-x^{2}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UHFhuJ6yJNDOcGH7Fz25/0LMmTq5Kv5XdxPzQ7kQWVs0qkEp2JxW0AnDoDp9iR0M3da3b9zhsFMhk04lIzK4SXWD310L1+P2yDQQfMEhENFy81jD1KXXftuJY668njgtdXOcpvTQWsvsfZ1iYKPP8umIuOEdZZHc3tYsORarla9N5Aod6Hr1Lp2e/29KhSgU4B8pwMJ96BND7tT5VRCYJBGly56NFXXshaZvAmAYBdtAE87xjAR3QcnJztTQT24W7LH0A7WZfDlyjdTEyiEO+q/kFmfrA/H8hCThX0HWQg0=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{-2x^{2}+4}{\\sqrt{4-x^{2}}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }