(\partial)/(\partial x)(2sqrt(x*y)-1/2 x^2)

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(2\\sqrt{x\\cdot\\:y}-\\frac{1}{2}x^{2}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(2\\sqrt{x\\cdot y}-\\frac{1}{2}x^{2})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "\\sqrt{\\frac{y}{x}}-x", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(2\\sqrt{xy}-\\frac{1}{2}x^{2}\\right)=\\sqrt{\\frac{y}{x}}-x$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(2\\sqrt{xy}-\\frac{1}{2}x^{2}\\right)", "steps": [ { "type": "interim", "title": "Simplify $$2\\sqrt{xy}-\\frac{1}{2}x^{2}:{\\quad}2\\sqrt{xy}-\\frac{x^{2}}{2}$$", "input": "2\\sqrt{xy}-\\frac{1}{2}x^{2}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{2}x^{2}=\\frac{x^{2}}{2}$$", "input": "\\frac{1}{2}x^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x^{2}}{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79noABpxc4IZFb3O0CFaPASdVBn2NNCFZqg4ZoVh6UwqjkVi15I8rBefLi4Iyt2wrqh2FHwNgeYX+xRFflBDUAT/L0MoYg+CUn6oyL3EO7Yp+kTAQ0wijjZT3CzKzoPHUjWa5dKN7qgdYIdTn58c30Y8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "result": "=2\\sqrt{xy}-\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(2\\sqrt{xy}-\\frac{x^{2}}{2}\\right)" }, { "type": "step", "primary": "Treat $$y\\:$$as a constant" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(2\\sqrt{xy}\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x^{2}}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(2\\sqrt{xy}\\right)=\\sqrt{\\frac{y}{x}}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(2\\sqrt{xy}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{\\partial\\:}{\\partial\\:x}\\left(\\sqrt{xy}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{xy}}\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\sqrt{xy}\\right)", "result": "=\\frac{1}{2\\sqrt{xy}}\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=xy$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(\\sqrt{u}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(\\sqrt{u}\\right)=\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\left(\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(u^{\\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": "=\\frac{1}{2}u^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}u^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{u}}$$", "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": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=u^{-\\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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKe0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjMvOxDqXzE3/CFO0TFmffHAH2kDe5DGYTz3TrPquGdIhyukSOA/1RgMKO0TMhInPOMabdUggEogUL9RT7PNKh0VQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}u^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "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": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{u}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{u}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JOPQ2g2GS9EQptV8nckZSrH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJv2RkT96g5Q5jVbn5fyeQzwB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2rvsVWktCxa7XSYzIK90x3+aTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{u}}\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)" }, { "type": "step", "primary": "Substitute back $$u=xy$$", "result": "=\\frac{1}{2\\sqrt{xy}}\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAm8pe5pZooY9F76lMXc+eW7hUG3ehOs1lVTUNFnEOFcstTaRI+MlBUxEaHqiUUS2BwT4mfudsQdl9XEOhHRuSEDhCJTF6MA1EfdSpf0hOXKBWRI6gtQV6C3py6o6agV4be2CNnwanyZsA0WsLyuo+purhXHi/NHF/lcJBH8m03Z+YEfd615wJbmsw+mfnotEc9SBv6izheLVUKQ/emokAUymtoIVIiI8N9ZX5PJNj61yLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)=y$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(xy\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=y\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=y\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkCHrPsxDHNAfe2TPFce/rinFjOV6V4e2DrBKqW1EhFu5KCf/kU7OLqVO3RHXJ9Wv0OG38IleojCyebAtZy+3Tm9JiwEB0ZXmaMqMWNNpbCrjG1uBlSxEBevFU4pf3SPbWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{2\\sqrt{xy}}y" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{1}{2\\sqrt{xy}}y:{\\quad}\\sqrt{\\frac{y}{x}}$$", "input": "2\\cdot\\:\\frac{1}{2\\sqrt{xy}}y", "result": "=\\sqrt{\\frac{y}{x}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2y}{2\\sqrt{xy}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1\\cdot\\:y}{\\sqrt{xy}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:y=y$$", "result": "=\\frac{y}{\\sqrt{yx}}" }, { "type": "interim", "title": "Factor $$\\sqrt{yx}:{\\quad}\\sqrt{y}\\sqrt{x}$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "result": "=\\sqrt{y}\\sqrt{x}" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{y}{\\sqrt{y}\\sqrt{x}}" }, { "type": "interim", "title": "Cancel $$\\frac{y}{\\sqrt{x}\\sqrt{y}}:{\\quad}\\frac{\\sqrt{y}}{\\sqrt{x}}$$", "input": "\\frac{y}{\\sqrt{x}\\sqrt{y}}", "result": "=\\frac{\\sqrt{y}}{\\sqrt{x}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{y}=y^{\\frac{1}{2}}$$" ], "result": "=\\frac{y}{y^{\\frac{1}{2}}\\sqrt{x}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{y^{1}}{y^{\\frac{1}{2}}}=y^{1-\\frac{1}{2}}$$" ], "result": "=\\frac{y^{1-\\frac{1}{2}}}{\\sqrt{x}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=\\frac{y^{\\frac{1}{2}}}{\\sqrt{x}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$y^{\\frac{1}{2}}=\\sqrt{y}$$" ], "result": "=\\frac{\\sqrt{y}}{\\sqrt{x}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjEP/+4rMmkKtXP0lOp7OSFKUDIohh31lRymt/QFLzzNCUCWbkwGOY7PqKo3U/JLJegT4HcZpSp1R2uDAGeKel5ecH99gJxrutAMqpNKf6LMo3oe/oyhMy2+1TQhDBd2f5CDbprbpsc66sdPy4iJzM9O3DnP75oWhpA2FStW0p16xbo02WPWFMMXxAmyL6zOIw==" } }, { "type": "step", "primary": "Combine same powers : $$\\frac{\\sqrt{x}}{\\sqrt{y}}=\\sqrt{\\frac{x}{y}}$$", "result": "=\\sqrt{\\frac{y}{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvimRI6gtQV6C3py6o6agV4bdLEExALrl6PWSpc3fDbWRbQslTDKxOR/6J+ZOGvUcaui8GzCD1T8ndxnMSCPgnUJAB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC60l0D77ht0SUAIhkGN1+oC0/h1b/xhPxfsYnYXov3MvMgmFidzLn8qAVzHC0VZJUA==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x^{2}}{2}\\right)=x$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x^{2}}{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\frac{1}{2}\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{2}\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\cdot\\:2x^{2-1}:{\\quad}x$$", "input": "\\frac{1}{2}\\cdot\\:2x^{2-1}", "result": "=x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x^{2-1}}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1\\cdot\\:x^{2-1}" }, { "type": "interim", "title": "$$x^{2-1}=x$$", "input": "x^{2-1}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$2-1=1$$", "result": "=x^{1}" }, { "type": "step", "primary": "Apply rule $$a^{1}=a$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MrFmiUZwycbqN9oVvvuBIAOfOVs9mPIqDLV5QIWwt3kj25an51zZEW52DIN7IXDtXqs+GBzatrTYCfd+mmvSWrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8RYahDgkU0qhrtp0v+o3QgzehkKrn0era9rz8TlL+x/vjFJYCq5U0+2hzbvgLJypfoEFMST8lDZxn1Yq5HMKVTtAb51BceE+a5ppiVMbLOTZm0OUp9nwmzb09ceQZ5nHJLqPFeMAv2RMLBzqY1FeCKw=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\sqrt{\\frac{y}{x}}-x" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }