derivative of (6u^2)/((u^2+u)^3)

{ "query": { "display": "derivative of $$\\frac{6u^{2}}{\\left(u^{2}+u\\right)^{3}}$$", "symbolab_question": "PRE_CALC#derivative \\frac{6u^{2}}{(u^{2}+u)^{3}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{6(-4u-1)}{u^{2}(u+1)^{4}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\frac{6u^{2}}{\\left(u^{2}+u\\right)^{3}}\\right)=\\frac{6\\left(-4u-1\\right)}{u^{2}\\left(u+1\\right)^{4}}$$", "input": "\\frac{d}{du}\\left(\\frac{6u^{2}}{\\left(u^{2}+u\\right)^{3}}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{du}\\left(\\frac{u^{2}}{\\left(u^{2}+u\\right)^{3}}\\right)" }, { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=6\\cdot\\:\\frac{\\frac{d}{du}\\left(u^{2}\\right)\\left(u^{2}+u\\right)^{3}-\\frac{d}{du}\\left(\\left(u^{2}+u\\right)^{3}\\right)u^{2}}{\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\left(u^{2}+u\\right)^{3}\\right)=3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)$$", "input": "\\frac{d}{du}\\left(\\left(u^{2}+u\\right)^{3}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(u^{2}+u\\right)^{2}\\frac{d}{du}\\left(u^{2}+u\\right)$$", "input": "\\frac{d}{du}\\left(\\left(u^{2}+u\\right)^{3}\\right)", "result": "=3\\left(u^{2}+u\\right)^{2}\\frac{d}{du}\\left(u^{2}+u\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=v^{3},\\:\\:v=\\left(u^{2}+u\\right)$$" ], "result": "=\\frac{d}{dv}\\left(v^{3}\\right)\\frac{d}{du}\\left(\\left(u^{2}+u\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{3}\\right)=3v^{2}$$", "input": "\\frac{d}{dv}\\left(v^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3v^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3v^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqpXEbeKr37jRuilFzr8ZE6k3hxk9aCfAWodBRxXgUexYAsXL0SggoaWzn1E3qRqh/8//6/nV5O4fb8Xgwi7maoRk7nr9IDbDGcsZRPmsBYLeGPG1dVOtRmH2mekt/+Ztw==" } }, { "type": "step", "result": "=3v^{2}\\frac{d}{du}\\left(\\left(u^{2}+u\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$v=\\left(u^{2}+u\\right)$$", "result": "=3\\left(u^{2}+u\\right)^{2}\\frac{d}{du}\\left(u^{2}+u\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqe9RxJVG0OVqIUnx9dZLUlkI3N32iJxie2D57vJEYsgOiaLJuL5RxgumX0gNvT19bcYBvJrr0UVRZhze7mTrdr2t6NPpG5GYhBBzTu6UnPzNCbzY7ORYSX+D9Cuvn748CQ/KPbNdBHSBvu+ZmNdli6jeh7+jKEzLb7VNCEMF3Z/Maswa5oIagKQFOfrnKm5+Gbb/PxX3awut5KxRxyZf8A=" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}+u\\right)=2u+1$$", "input": "\\frac{d}{du}\\left(u^{2}+u\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{du}\\left(u^{2}\\right)+\\frac{du}{du}" }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "interim", "title": "$$\\frac{du}{du}=1$$", "input": "\\frac{du}{du}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{du}{du}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYipXfuvZQYcWZ3RXZmhAT9ljqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIfqIOlxNXEONDm3M0PlIv9pOXvV+QvzGT1U5/bJzrRe1" } }, { "type": "step", "result": "=2u+1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=6\\cdot\\:\\frac{2u\\left(u^{2}+u\\right)^{3}-3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)u^{2}}{\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$6\\cdot\\:\\frac{2u\\left(u^{2}+u\\right)^{3}-3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)u^{2}}{\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}}:{\\quad}\\frac{6\\left(-4u-1\\right)}{u^{2}\\left(u+1\\right)^{4}}$$", "input": "6\\cdot\\:\\frac{2u\\left(u^{2}+u\\right)^{3}-3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)u^{2}}{\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}}", "result": "=\\frac{6\\left(-4u-1\\right)}{u^{2}\\left(u+1\\right)^{4}}", "steps": [ { "type": "interim", "title": "$$\\frac{2u\\left(u^{2}+u\\right)^{3}-3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)u^{2}}{\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}}=\\frac{-4u-1}{u^{2}\\left(u+1\\right)^{4}}$$", "input": "\\frac{2u\\left(u^{2}+u\\right)^{3}-3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)u^{2}}{\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}}", "steps": [ { "type": "interim", "title": "Factor $$2u\\left(u^{2}+u\\right)^{3}-3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)u^{2}:{\\quad}u^{2}\\left(u^{2}+u\\right)^{2}\\left(-4u-1\\right)$$", "input": "2u\\left(u^{2}+u\\right)^{3}-3\\left(u^{2}+u\\right)^{2}\\left(2u+1\\right)u^{2}", "result": "=\\frac{u^{2}\\left(u^{2}+u\\right)^{2}\\left(-4u-1\\right)}{\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\left(u^{2}+u\\right)^{3}=\\left(u^{2}+u\\right)^{2}\\left(u^{2}+u\\right),\\:u^{2}=uu$$" ], "result": "=2u\\left(u^{2}+u\\right)^{2}\\left(u^{2}+u\\right)-3\\left(uu+u\\right)^{2}\\left(1+2u\\right)uu", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u\\left(u^{2}+u\\right)^{2}$$", "result": "=u\\left(u^{2}+u\\right)^{2}\\left(2\\left(u^{2}+u\\right)-3\\left(1+2u\\right)u\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$-3u\\left(2u+1\\right)+2\\left(u^{2}+u\\right):{\\quad}u\\left(-4u-1\\right)$$", "input": "2\\left(u^{2}+u\\right)-3\\left(1+2u\\right)u", "result": "=uu\\left(-4u-1\\right)\\left(u^{2}+u\\right)^{2}", "steps": [ { "type": "interim", "title": "Factor $$u^{2}+u:{\\quad}u\\left(u+1\\right)$$", "input": "u^{2}+u", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{2}=uu$$" ], "result": "=uu+u", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u$$", "result": "=u\\left(u+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=2u\\left(u+1\\right)-3u\\left(2u+1\\right)" }, { "type": "step", "primary": "Factor out common term $$u$$", "result": "=u\\left(2\\left(1+u\\right)-3\\left(1+2u\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Expand $$2\\left(u+1\\right)-3\\left(2u+1\\right):{\\quad}-4u-1$$", "input": "2\\left(1+u\\right)-3\\left(1+2u\\right)", "result": "=u\\left(-4u-1\\right)", "steps": [ { "type": "interim", "title": "Expand $$2\\left(1+u\\right):{\\quad}2+2u$$", "input": "2\\left(1+u\\right)", "result": "=2+2u-3\\left(1+2u\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=2,\\:b=1,\\:c=u$$" ], "result": "=2\\cdot\\:1+2u", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2+2u" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qLXklnEPd0lZ9MBbYzqPgHWD310L1+P2yDQQfMEhENE6wizQyrmJ3gsPT3K0aDlO1sD7NfhsPe7eDHrmjY0mE4CVqp+SQd1u5obYIE9RuLOwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Expand $$-3\\left(1+2u\\right):{\\quad}-3-6u$$", "input": "-3\\left(1+2u\\right)", "result": "=2+2u-3-6u", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-3,\\:b=1,\\:c=2u$$" ], "result": "=-3\\cdot\\:1+\\left(-3\\right)\\cdot\\:2u", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-3\\cdot\\:1-3\\cdot\\:2u" }, { "type": "interim", "title": "Simplify $$-3\\cdot\\:1-3\\cdot\\:2u:{\\quad}-3-6u$$", "input": "-3\\cdot\\:1-3\\cdot\\:2u", "result": "=-3-6u", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:1=3$$", "result": "=-3-3\\cdot\\:2u" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=-3-6u" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mQ4niXg+8tBBD4OmI7unbc0ag8T1MwTer44+aCS/ZFB26T7HFMFSXLbzAfOuJsiW72wZm7kDUxdE6YSmfEbr2mPWFmYbm6gt7L0gsaIfTfVUWGAMtFqtuJ7Q6C+zgrZF" } }, { "type": "interim", "title": "Simplify $$2+2u-3-6u:{\\quad}-4u-1$$", "input": "2+2u-3-6u", "result": "=-4u-1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2u-6u+2-3" }, { "type": "step", "primary": "Add similar elements: $$2u-6u=-4u$$", "result": "=-4u+2-3" }, { "type": "step", "primary": "Add/Subtract the numbers: $$2-3=-1$$", "result": "=-4u-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l0WDayZ0UQ3BMv5p9klPwgCWKUbvV6WK3fDUgFtg3Q9bqMsGOawvMhL8mlXS9Xs35UznuILidFkoxGMktjUdr4Jy8vaNLPDVWVFwZfiRrqqkFn7p+4wXjXm6SGcWHQrP4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Refine", "result": "=u^{2}\\left(-4u-1\\right)\\left(u^{2}+u\\right)^{2}" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}:{\\quad}\\left(u^{2}+u\\right)^{6}$$", "input": "\\left(\\left(u^{2}+u\\right)^{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(u^{2}+u\\right)^{3\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=\\left(u^{2}+u\\right)^{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ivL+CCI8u30tFi9Z3YQ2/uiEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPIkavUyuXNRP0r+J55AruPX2RLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9Bo9bC/iyqfoHpllvuYvuxmtTVmd09XElMYIMJjAj0C4=" } }, { "type": "step", "result": "=\\frac{u^{2}\\left(-4u-1\\right)\\left(u^{2}+u\\right)^{2}}{\\left(u^{2}+u\\right)^{6}}" }, { "type": "interim", "title": "Cancel $$\\frac{u^{2}\\left(u^{2}+u\\right)^{2}\\left(-4u-1\\right)}{\\left(u^{2}+u\\right)^{6}}:{\\quad}\\frac{u^{2}\\left(-4u-1\\right)}{\\left(u^{2}+u\\right)^{4}}$$", "input": "\\frac{u^{2}\\left(u^{2}+u\\right)^{2}\\left(-4u-1\\right)}{\\left(u^{2}+u\\right)^{6}}", "result": "=\\frac{u^{2}\\left(-4u-1\\right)}{\\left(u^{2}+u\\right)^{4}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{\\left(u^{2}+u\\right)^{2}}{\\left(u^{2}+u\\right)^{6}}=\\frac{1}{\\left(u^{2}+u\\right)^{6-2}}$$" ], "result": "=\\frac{u^{2}\\left(-4u-1\\right)}{\\left(u^{2}+u\\right)^{6-2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$6-2=4$$", "result": "=\\frac{u^{2}\\left(-4u-1\\right)}{\\left(u^{2}+u\\right)^{4}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkIPccR4V8ZcFTbMymycFkGGgAaLR6fncErWkWzcvgP8ZIPElQpgkHlSBSM0dkGj880ag8T1MwTer44+aCS/ZFA9Gr3Gyxsl9UAzoUkd90d98t00ebE7DKccnNa30J3DJ7JJKpAhxZ4AXs165kJN/YV6pfF1z6umzUJTJvt+ojYZ3Rhao/hEKgGYcV+JSwxeA4bO+1UU6+/4SCPFR01NyrPy3TR5sTsMpxyc1rfQncMnGVR0+M0LKDsBpgSwsbLMoA==" } }, { "type": "interim", "title": "Factor $$\\left(u^{2}+u\\right)^{4}:{\\quad}u^{4}\\left(u+1\\right)^{4}$$", "input": "\\left(u^{2}+u\\right)^{4}", "result": "=\\frac{u^{2}\\left(-4u-1\\right)}{u^{4}\\left(u+1\\right)^{4}}", "steps": [ { "type": "interim", "title": "Factor $$\\left(u^{2}+u\\right)^{4}:{\\quad}u^{4}\\left(u+1\\right)^{4}$$", "steps": [ { "type": "interim", "title": "Factor $$u^{2}+u:{\\quad}u\\left(u+1\\right)$$", "input": "u^{2}+u", "steps": [ { "type": "interim", "title": "Factor out common term $$u:{\\quad}u\\left(u+1\\right)$$", "input": "u^{2}+u", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{2}=uu$$" ], "result": "=uu+u", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u$$", "result": "=u\\left(u+1\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=u\\left(u+1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\left(u\\left(u+1\\right)\\right)^{4}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(ab\\right)^n=a^{n}b^{n}$$", "result": "=u^{4}\\left(u+1\\right)^{4}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=u^{4}\\left(u+1\\right)^{4}" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Cancel $$\\frac{u^{2}\\left(-4u-1\\right)}{u^{4}\\left(u+1\\right)^{4}}:{\\quad}\\frac{-4u-1}{u^{2}\\left(u+1\\right)^{4}}$$", "input": "\\frac{u^{2}\\left(-4u-1\\right)}{u^{4}\\left(u+1\\right)^{4}}", "result": "=\\frac{-4u-1}{u^{2}\\left(u+1\\right)^{4}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{u^{2}}{u^{4}}=\\frac{1}{u^{4-2}}$$" ], "result": "=\\frac{-4u-1}{u^{4-2}\\left(u+1\\right)^{4}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$4-2=2$$", "result": "=\\frac{-4u-1}{u^{2}\\left(u+1\\right)^{4}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlLlXvJJIHbB13oaT4uuzRBoMy8ttT3hvCuwT6/q4AylVdNK6b/PmQukzNdLVQkh7GCeK3dtjkfH1uaY9WqMYsBN+EMVptjSpIlCoTsea/5CjIbtoVyuHBmtvJfLTdQAFJzRQCb3t7dl0WzvSMzsBj7zXB/plvJ2hdDMbGWW2EJVnzhzrVSAIZ7CHWrBt3Mc8x6yk35TRQX/XkxnVI8y2m0=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cT+PQNKFPdBoRdbpyr0eu6o7v/riQ7cQfPQhuDVJiC0FTvDl6Wgaw1HnvamAjXy1+PzXd7D8xQqhFJY4IbxQ6jsFYHTBUHP4viT7ukZsvgF1g99dC9fj9sg0EHzBIRDRaG7ux4DWHicmZh9MuiFE8F3XWgn4rv/EXyyiW/yZFIwB9pA3uQxmE8906z6rhnSI7POKqrwplL+994zSxWg0hqo7v/riQ7cQfPQhuDVJiC0FTvDl6Wgaw1HnvamAjXy1+PzXd7D8xQqhFJY4IbxQ6gygGRcsV5jTf6EYXQzAhthjZ7NTXXaD+1LJSMRkOOk4RP4FDdYk5iyRfr6bMYNDKA==" } }, { "type": "step", "result": "=6\\cdot\\:\\frac{-4u-1}{u^{2}\\left(u+1\\right)^{4}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(-4u-1\\right)\\cdot\\:6}{u^{2}\\left(u+1\\right)^{4}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wHsyNILr7YqAxQ7kxdsiXmJncPyHnox7xCy4kJfriWILnp0/PagFFXvMzqiq0kdQ38j8mccj8lIrnvmeoq23F3oiTM4AUGHvOTBL8JRcjB8AlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJq6Egfh5EUTIgkgAkrCTfE7pQtbrsGhvJQwFQ9/4B9/Qv1AB4nlpCjp4ECP9OlpADJjcBIL5pmo83UMFZRSzJsVuAqeycJR4bH/r8QyGOuhJ2gQ2wtn8F3B5tcBsuuDSyyEJQ4lHUN6HQE581omzir5UqxfHCMasbC85kK9JUhAzVjzNwPuCdGKH2h6gbQf036V11Qc2JhHGxcZHALGOm/I=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "u", "plotRequest": "y=\\frac{6(-4u-1)}{u^{2}(u+1)^{4}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }