derivative of f(x)=x^8sqrt(5-3x)

{ "query": { "display": "derivative of $$f\\left(x\\right)=x^{8}\\sqrt{5-3x}$$", "symbolab_question": "PRE_CALC#derivative f(x)=x^{8}\\sqrt{5-3x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{-51x^{8}+80x^{7}}{2\\sqrt{5-3x}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{8}\\sqrt{5-3x}\\right)=\\frac{-51x^{8}+80x^{7}}{2\\sqrt{5-3x}}$$", "input": "\\frac{d}{dx}\\left(x^{8}\\sqrt{5-3x}\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x^{8},\\:g=\\sqrt{5-3x}$$" ], "result": "=\\frac{d}{dx}\\left(x^{8}\\right)\\sqrt{5-3x}+\\frac{d}{dx}\\left(\\sqrt{5-3x}\\right)x^{8}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{8}\\right)=8x^{7}$$", "input": "\\frac{d}{dx}\\left(x^{8}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=8x^{8-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8x^{7}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrGwFOVnKdpmGzntotIijwGk3hxk9aCfAWodBRxXgUex2qP3yLWbwuf7ykux/+ljNv8//6/nV5O4fb8Xgwi7maqO95Is7YBdcRqXmTH8euc2S/vtDNB6GWteJZBSm7X14g==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{5-3x}\\right)=-\\frac{3}{2\\sqrt{5-3x}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{5-3x}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{5-3x}}\\frac{d}{dx}\\left(5-3x\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{5-3x}\\right)", "result": "=\\frac{1}{2\\sqrt{5-3x}}\\frac{d}{dx}\\left(5-3x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=5-3x$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(5-3x\\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": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{du}\\left(u^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "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{d}{dx}\\left(5-3x\\right)" }, { "type": "step", "primary": "Substitute back $$u=5-3x$$", "result": "=\\frac{1}{2\\sqrt{5-3x}}\\frac{d}{dx}\\left(5-3x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtuiB8si2Rzy29SaC+ZRTlQB562iL4I4GzHU2vCEsUhM1NpEj4yUFTERoeqJRRLYHBPiZ+52xB2X1cQ6EdG5IQOEIlMXowDUR91Kl/SE5coFeeR1OdpCoc1m9nQe8Y8O0mm86F16QV7LwbyCJvoM1LVGJftth69Dsi14RaP9crrmStvD8GKDNJJN8qiu0XRcVZAYAvbRKqkRVDaDBHkSBtDwYHkfmHHTze3JXk3LjiJp" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5-3x\\right)=-3$$", "input": "\\frac{d}{dx}\\left(5-3x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(5\\right)-\\frac{d}{dx}\\left(3x\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5\\right)=0$$", "input": "\\frac{d}{dx}\\left(5\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmXEh6/dOKVl5+UiJ6t4qwxJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTvz/OzRy6l5fd6++0L3aMbw" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x\\right)=3$$", "input": "\\frac{d}{dx}\\left(3x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsUnAVaDXCLQOuynYB+k3fTZGku9zFkxwe1dTH8vycb9BbqPJ4kl+ElAajU+EBTcW1NbbqpyK7JQEZdATEJR51i3CwF9WgU8/+rFW242YnYD" } }, { "type": "step", "result": "=0-3" }, { "type": "step", "primary": "Simplify", "result": "=-3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{5-3x}}\\left(-3\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\sqrt{5-3x}}\\left(-3\\right):{\\quad}-\\frac{3}{2\\sqrt{5-3x}}$$", "input": "\\frac{1}{2\\sqrt{5-3x}}\\left(-3\\right)", "result": "=-\\frac{3}{2\\sqrt{5-3x}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2\\sqrt{5-3x}}\\cdot\\:3" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:3}{2\\sqrt{5-3x}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=-\\frac{3}{2\\sqrt{-3x+5}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13C5pDM1OsLdjd5Ir+Awl+9pVV00rpv8+ZC6TM10tVCSHshgFkMSyGhkcJufVqsMeq9ACqtCxoEdJkzHBdj1Wzq8yzC+0FcPHGky5+f+dHTyvCixpjZyz1hn/Tx95aZo6CSFjbCAuoanhwjo/9bMp/72k/j1SwP5vhsJN1hTPkelsprWWW8bmUThqRx9RONaSjag==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=8x^{7}\\sqrt{5-3x}+\\left(-\\frac{3}{2\\sqrt{5-3x}}\\right)x^{8}" }, { "type": "interim", "title": "Simplify $$8x^{7}\\sqrt{5-3x}+\\left(-\\frac{3}{2\\sqrt{5-3x}}\\right)x^{8}:{\\quad}\\frac{-51x^{8}+80x^{7}}{2\\sqrt{5-3x}}$$", "input": "8x^{7}\\sqrt{5-3x}+\\left(-\\frac{3}{2\\sqrt{5-3x}}\\right)x^{8}", "result": "=\\frac{-51x^{8}+80x^{7}}{2\\sqrt{5-3x}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=8x^{7}\\sqrt{5-3x}-\\frac{3}{2\\sqrt{5-3x}}x^{8}" }, { "type": "interim", "title": "Multiply $$\\frac{3}{2\\sqrt{5-3x}}x^{8}\\::{\\quad}\\frac{3x^{8}}{2\\sqrt{-3x+5}}$$", "input": "\\frac{3}{2\\sqrt{5-3x}}x^{8}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3x^{8}}{2\\sqrt{5-3x}}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=8x^{7}\\sqrt{-3x+5}-\\frac{3x^{8}}{2\\sqrt{-3x+5}}" }, { "type": "step", "primary": "Convert element to fraction: $$8x^{7}\\sqrt{-3x+5}=\\frac{8x^{7}\\sqrt{5-3x}\\cdot\\:2\\sqrt{5-3x}}{2\\sqrt{5-3x}}$$", "result": "=-\\frac{3x^{8}}{2\\sqrt{5-3x}}+\\frac{8x^{7}\\sqrt{5-3x}\\cdot\\:2\\sqrt{5-3x}}{2\\sqrt{5-3x}}" }, { "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{-3x^{8}+8x^{7}\\sqrt{5-3x}\\cdot\\:2\\sqrt{5-3x}}{2\\sqrt{5-3x}}" }, { "type": "interim", "title": "$$-3x^{8}+8x^{7}\\sqrt{5-3x}\\cdot\\:2\\sqrt{5-3x}=-3x^{8}+16x^{7}\\left(5-3x\\right)$$", "input": "-3x^{8}+8x^{7}\\sqrt{5-3x}\\cdot\\:2\\sqrt{5-3x}", "steps": [ { "type": "interim", "title": "$$8x^{7}\\sqrt{5-3x}\\cdot\\:2\\sqrt{5-3x}=16x^{7}\\left(5-3x\\right)$$", "input": "8x^{7}\\sqrt{5-3x}\\cdot\\:2\\sqrt{5-3x}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:2=16$$", "result": "=16x^{7}\\sqrt{-3x+5}\\sqrt{-3x+5}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{-3x+5}\\sqrt{-3x+5}=5-3x$$" ], "result": "=16x^{7}\\left(-3x+5\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70rfKRht1jS4JhqVnifXwDNwbR2m3LH/9FGJvnTYxZdA2R4JXe2oN+QvDiQq5u/WidYPfXQvX4/bINBB8wSEQ0boWijuGStNON8mWCXU7s1tEkFyRLR03M11bGpdZyLWisiwU9ry+6SKUt0uXNSvnBrNbboAuZbW4ZbYYhsqcvMjFJp+Hd/ke3PqqoTq6CgQrzOjpTpyoG/uYos09D0eh4w==" } }, { "type": "step", "result": "=-3x^{8}+16x^{7}\\left(-3x+5\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzhhe7UOtYCUB0D5DuC4DEhS76sVM57z8PdWpIH9tFXZScz5iX4TRahQ5GK7/I6NAJYpRu9XpYrd8NSAW2DdD06SsUW29uMFscSW3cunWVmup4g19A5y4ztDs2PnQqEpRJBckS0dNzNdWxqXWci1oiod1jGnjaZyAxE3Mfohv6DQOPwwJqBdy9j2UpDI5C1m7KsNGIuGlJr91GGZbF+9FmKCz4SO5sGMYs5At4ykGTPM6OlOnKgb+5iizT0PR6Hj" } }, { "type": "step", "result": "=\\frac{-3x^{8}+16x^{7}\\left(-3x+5\\right)}{2\\sqrt{-3x+5}}" }, { "type": "interim", "title": "Expand $$-3x^{8}+16x^{7}\\left(5-3x\\right):{\\quad}-51x^{8}+80x^{7}$$", "input": "-3x^{8}+16x^{7}\\left(5-3x\\right)", "result": "=\\frac{-51x^{8}+80x^{7}}{2\\sqrt{-3x+5}}", "steps": [ { "type": "interim", "title": "Expand $$16x^{7}\\left(5-3x\\right):{\\quad}80x^{7}-48x^{8}$$", "input": "16x^{7}\\left(5-3x\\right)", "result": "=-3x^{8}+80x^{7}-48x^{8}", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=16x^{7},\\:b=5,\\:c=3x$$" ], "result": "=16x^{7}\\cdot\\:5-16x^{7}\\cdot\\:3x", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=16\\cdot\\:5x^{7}-16\\cdot\\:3x^{7}x" }, { "type": "interim", "title": "Simplify $$16\\cdot\\:5x^{7}-16\\cdot\\:3x^{7}x:{\\quad}80x^{7}-48x^{8}$$", "input": "16\\cdot\\:5x^{7}-16\\cdot\\:3x^{7}x", "result": "=80x^{7}-48x^{8}", "steps": [ { "type": "interim", "title": "$$16\\cdot\\:5x^{7}=80x^{7}$$", "input": "16\\cdot\\:5x^{7}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:5=80$$", "result": "=80x^{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oa3UcTQ1JODhdgc5K/ePLXyRHuGw7+tM5METTDj6vVFVOG9wm4cbG+Oc50zif9r9GYWIwq7tMvHsgtxnvOBtjPEKwPomHX0Hj8mZRbHgDhqepZ5NC3dR2tBo+FunQM6D" } }, { "type": "interim", "title": "$$16\\cdot\\:3x^{7}x=48x^{8}$$", "input": "16\\cdot\\:3x^{7}x", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:3=48$$", "result": "=48x^{7}x" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{7}x=\\:x^{7+1}$$" ], "result": "=48x^{7+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$7+1=8$$", "result": "=48x^{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s734RPgUtPOjumKfqlWu/E2PxI6rtl1NtcN/hmtlZD7xY8LIDuLGMJ18Nh77Jpzso6l9OccPYZS6XnwzWS/KVmVl2Roun23GnmQ6pLpfFpxtiI8fmXHJeMJ2n01SlYnKj0JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=80x^{7}-48x^{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75yqW9zMJDZ/q3tzPR1Nn6y061ljBSPJeENOw2efoSWvhjXA25QZrtgzx4JuGFD/0ezvI1vLdpFNUL/jbMKrFkYEFMST8lDZxn1Yq5HMKVTt5j0JkVh2FYUT8H0AbfqOAlImLVUMLq3s3jjR+qlm0Lw==" } }, { "type": "interim", "title": "Simplify $$-3x^{8}+80x^{7}-48x^{8}:{\\quad}-51x^{8}+80x^{7}$$", "input": "-3x^{8}+80x^{7}-48x^{8}", "result": "=-51x^{8}+80x^{7}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-3x^{8}-48x^{8}+80x^{7}" }, { "type": "step", "primary": "Add similar elements: $$-3x^{8}-48x^{8}=-51x^{8}$$", "result": "=-51x^{8}+80x^{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WibMeWR71hMQRFY8qkig58az+7uDWdD0Z4a8KvMafusJQJZuTAY5js+oqjdT8kslFOaOub7Tx3oYMuiO4WUhuYX0K5CYImK/dpECBv0CZ44ezFilETfuNygjs0XPkV2hZVvptnwEsu5TSPIVKo2Eu1i6FEPyPyFtanWsXCLWKTc=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70rfKRht1jS4JhqVnifXwDHiEWdBzcMp+7J02liP9h52XEP6E8UIMc+VNgkOrgibc/InOClqqWtiH9h7DcJ8ILnCQoYlYQ8U+Tfyx0kyzI8jg1aMB+XDtDRnf60+pbfvlna5EUVPd6EkY94pWrYwmPxMgLF5ZcnOvRwXWatDF/7Hwt9LEn7QCBUukJKctfSJK85vD08JljwYZGcDAvUWmu7uRhTZ8nh1+grcbEmOObKF3VvvGUdoe+zQ71kfLAj5XVXJCPYMOOkQjb4JtlKB7bLCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{-51x^{8}+80x^{7}}{2\\sqrt{5-3x}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }