(\partial)/(\partial x)(6y^2sqrt(x))

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(6y^{2}\\sqrt{x}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(6y^{2}\\sqrt{x})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "\\frac{3y^{2}}{\\sqrt{x}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(6y^{2}\\sqrt{x}\\right)=\\frac{3y^{2}}{\\sqrt{x}}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(6y^{2}\\sqrt{x}\\right)", "steps": [ { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6y^{2}\\frac{\\partial\\:}{\\partial\\:x}\\left(\\sqrt{x}\\right)" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=6y^{2}\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=6y^{2}\\frac{1}{2}x^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$6y^{2}\\frac{1}{2}x^{\\frac{1}{2}-1}:{\\quad}\\frac{3y^{2}}{\\sqrt{x}}$$", "input": "6y^{2}\\frac{1}{2}x^{\\frac{1}{2}-1}", "result": "=\\frac{3y^{2}}{\\sqrt{x}}", "steps": [ { "type": "interim", "title": "$$x^{\\frac{1}{2}-1}=x^{-\\frac{1}{2}}$$", "input": "x^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=x^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qijEBDcyPMwV4Y1jeiGyoO0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjQYCY06ctBCI/puUxKEtzAQH2kDe5DGYTz3TrPquGdIjtHZXPNLHlLyai31n5HH4G6M8osviUPEkWv33aMbZrSFQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=6\\cdot\\:\\frac{1}{2}y^{2}x^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{x}}$$" ], "result": "=6\\cdot\\:\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{x}}y^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{1\\cdot\\:1\\cdot\\:6y^{2}}{2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1\\cdot\\:6=6$$", "result": "=\\frac{6y^{2}}{2\\sqrt{x}}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{2}=3$$", "result": "=\\frac{3y^{2}}{\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73h3lYgcjAIniPCqwG08uuEQ2tnn4MvCupPKnOdfx5iu2oVwn9rqBac7JV/bk/6m+q47vuWedXv2WUg94ER8IwcsvPm+oG99EJykovdJnkIl/CGrjd5Qc8wJrT39a5pl372wZm7kDUxdE6YSmfEbr2mN2QVk5Uy6JLEPS3CeYJ2GtV4LZMo8tVPAr16ZyS6QXv6jY2qcQmwXzGCGE7nZw2E5OQF0x1Pgvk6xils6k/ew=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }