derivative of 11sqrt(x)-8/x

{ "query": { "display": "derivative of $$11\\sqrt{x}-\\frac{8}{x}$$", "symbolab_question": "PRE_CALC#derivative 11\\sqrt{x}-\\frac{8}{x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(11\\sqrt{x}-\\frac{8}{x}\\right)=\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}$$", "input": "\\frac{d}{dx}\\left(11\\sqrt{x}-\\frac{8}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(11\\sqrt{x}\\right)-\\frac{d}{dx}\\left(\\frac{8}{x}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(11\\sqrt{x}\\right)=\\frac{11}{2\\sqrt{x}}$$", "input": "\\frac{d}{dx}\\left(11\\sqrt{x}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=11\\frac{d}{dx}\\left(\\sqrt{x}\\right)" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=11\\frac{d}{dx}\\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": "=11\\cdot\\:\\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 $$11\\cdot\\:\\frac{1}{2}x^{\\frac{1}{2}-1}:{\\quad}\\frac{11}{2\\sqrt{x}}$$", "input": "11\\cdot\\:\\frac{1}{2}x^{\\frac{1}{2}-1}", "result": "=\\frac{11}{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": "=11\\cdot\\:\\frac{1}{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": "=11\\cdot\\:\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{x}}", "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\\:11}{2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1\\cdot\\:11=11$$", "result": "=\\frac{11}{2\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HDQ7Td+D3XJM5ZO06G7zcLH91i0rf1KOlMObZ0nnT2UO2t9gaEaI1WfhIM6jfIfTA585Wz2Y8ioMtXlAhbC3eWzooD/7AJAJhDMGpBv2Wi69dagAbfI9PhRUxWGqYEgL72wZm7kDUxdE6YSmfEbr2nVRc8yFkYN/irVBM1Pk3g+aV/9ZUwjs8bIu0QqbY0rWRDa2efgy8K6k8qc51/HmK+BLOmoIw0CSGGQK+Tt+2gE=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{8}{x}\\right)=-\\frac{8}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{8}{x}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dx}\\left(\\frac{1}{x}\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=8\\frac{d}{dx}\\left(x^{-1}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=8\\left(-1\\cdot\\:x^{-1-1}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$8\\left(-1\\cdot\\:x^{-1-1}\\right):{\\quad}-\\frac{8}{x^{2}}$$", "input": "8\\left(-1\\cdot\\:x^{-1-1}\\right)", "result": "=-\\frac{8}{x^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-8\\cdot\\:1\\cdot\\:x^{-1-1}" }, { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-8\\cdot\\:1\\cdot\\:x^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-2}=\\frac{1}{x^{2}}$$" ], "result": "=8\\cdot\\:1\\cdot\\:\\frac{1}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-1\\cdot\\:\\frac{1\\cdot\\:8}{x^{2}}" }, { "type": "step", "primary": "Refine", "result": "=-\\frac{8}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UqwMsQnsQb6g5s8B/I9JJ4zkasBnmYoEPfqkRQoS3Airju+5Z51e/ZZSD3gRHwjBT9MQoWrnLsSMsJ0PVzCvAek+KKUPvS9SYIBRdKMJESmY3ASC+aZqPN1DBWUUsybFb4WqrZcpfwv77lK6oOJZupdMyhpKN+SVuICGnXtSpSo=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{11}{2\\sqrt{x}}-\\left(-\\frac{8}{x^{2}}\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{11}{2\\sqrt{x}}-\\left(-\\frac{8}{x^{2}}\\right):{\\quad}\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}$$", "input": "\\frac{11}{2\\sqrt{x}}-\\left(-\\frac{8}{x^{2}}\\right)", "result": "=\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{11}{2\\sqrt{x}}+\\frac{8}{x^{2}}" }, { "type": "interim", "title": "Least Common Multiplier of $$2\\sqrt{x},\\:x^{2}:{\\quad}2x^{2}\\sqrt{x}$$", "input": "2\\sqrt{x},\\:x^{2}", "steps": [ { "type": "definition", "title": "Lowest Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b\\:$$is the smallest multiplier that is divisible by both $$a$$ and $$b$$" }, { "type": "step", "primary": "Compute an expression comprised of factors that appear either in $$2\\sqrt{x}$$ or $$x^{2}$$", "result": "=2x^{2}\\sqrt{x}" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$2x^{2}\\sqrt{x}$$" }, { "type": "step", "primary": "For $$\\frac{11}{2\\sqrt{x}}:\\:$$multiply the denominator and numerator by $$x^{\\frac{3}{2}}\\sqrt{x}$$", "result": "\\frac{11}{2\\sqrt{x}}=\\frac{11x^{\\frac{3}{2}}\\sqrt{x}}{2\\sqrt{x}x^{\\frac{3}{2}}\\sqrt{x}}=\\frac{11x^{\\frac{3}{2}}\\sqrt{x}}{2x^{2}\\sqrt{x}}" }, { "type": "step", "primary": "For $$\\frac{8}{x^{2}}:\\:$$multiply the denominator and numerator by $$2\\sqrt{x}$$", "result": "\\frac{8}{x^{2}}=\\frac{8\\cdot\\:2\\sqrt{x}}{x^{2}\\cdot\\:2\\sqrt{x}}=\\frac{16\\sqrt{x}}{2x^{2}\\sqrt{x}}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{11x^{\\frac{3}{2}}\\sqrt{x}}{2x^{2}\\sqrt{x}}+\\frac{16\\sqrt{x}}{2x^{2}\\sqrt{x}}" }, { "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{11x^{\\frac{3}{2}}\\sqrt{x}+16\\sqrt{x}}{2x^{2}\\sqrt{x}}" }, { "type": "step", "primary": "Factor out common term $$\\sqrt{x}$$", "result": "=\\frac{\\sqrt{x}\\left(11x^{\\frac{3}{2}}+16\\right)}{2x^{2}\\sqrt{x}}", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{x}\\left(11x^{\\frac{3}{2}}+16\\right)}{2x^{2}\\sqrt{x}}:{\\quad}\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}$$", "input": "\\frac{\\sqrt{x}\\left(11x^{\\frac{3}{2}}+16\\right)}{2x^{2}\\sqrt{x}}", "result": "=\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{x}=x^{\\frac{1}{2}}$$" ], "result": "=\\frac{x^{\\frac{1}{2}}\\left(11x^{\\frac{3}{2}}+16\\right)}{2x^{2}\\sqrt{x}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{x^{\\frac{1}{2}}}{x^{2}}=\\frac{1}{x^{2-\\frac{1}{2}}}$$" ], "result": "=\\frac{11x^{\\frac{3}{2}}+16}{2\\sqrt{x}x^{-\\frac{1}{2}+2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnTeddJmvqWjp/5j66VZOiyMiY0fH9q5SYcRf0wsrTQwgY8OFfyYtUZ+4BSji/Jmh+y3UWvJjk/Fi3m3Em6kIMbO9hoabBmW/LmoxzzZUjp1WYwoULRjgmhXurvi/+Bj1yRmXj8NmyL15ctghULGXH6l1rBGn8tHQGjm2hHV3uwKJ3ZwtBIBxh5Pt81SdJBOSE3kCh3oevUunZ7/b0qFKBQPcjB9OuvXvbKc1NcHm8+dyTL0YFDvJj7ldKrJQ/BNI0ne6/daCCRTGNstA7TN4J85u8DSk/46d01fip71MJ/vialcV/dI5TH4fXyp+ncwuA==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pNtEGDnfHelNlXPW3XFp3dchhz9eKs1e4tlWB4w4P8vO4I674sk024BYIQPoD8atICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyciayme0CNTLuyLBixw/iXeODbuwHnCGQ4NbPrKmIr9giyyPjxfNCAH6m77xlMXdF8B9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC6GLemlEyM0AqQK8K5f57MjemkHLXggjk3ebZq4dEHgG2X2UGliGN4oOWjefA9JBSL8yD3hLQ33B7/8/LpbPE3o=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{11x^{\\frac{3}{2}}+16}{2x^{\\frac{3}{2}}\\sqrt{x}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }