(\partial)/(\partial x)(-8(x+y+z)^2)

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(-8\\left(x+y+z\\right)^{2}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(-8(x+y+z)^{2})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "-16(x+y+z)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(-8\\left(x+y+z\\right)^{2}\\right)=-16\\left(x+y+z\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(-8\\left(x+y+z\\right)^{2}\\right)", "steps": [ { "type": "step", "primary": "Treat $$y,\\:z\\:$$as constants" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-8\\frac{\\partial\\:}{\\partial\\:x}\\left(\\left(x+y+z\\right)^{2}\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\left(x+y+z\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(x+y+z\\right)$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\left(x+y+z\\right)^{2}\\right)", "result": "=2\\left(x+y+z\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(x+y+z\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=\\left(x+y+z\\right)$$" ], "result": "=\\frac{\\partial\\:}{\\partial\\:u}\\left(u^{2}\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(\\left(x+y+z\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:u}\\left(u^{2}\\right)=2u$$", "input": "\\frac{\\partial\\:}{\\partial\\:u}\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlk+sT8PdKjX3IOZNAv4DdeHI5S0StY1FdtOqqOPr0Te9NVo9MACLuml/Lpxq9I00hkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089tuZ+dDFI37AdjP7AT1e0ClUFJjxir5UHSHqDvA4SbOG" } }, { "type": "step", "result": "=2u\\frac{\\partial\\:}{\\partial\\:x}\\left(\\left(x+y+z\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\left(x+y+z\\right)$$", "result": "=2\\left(x+y+z\\right)\\frac{\\partial\\:}{\\partial\\:x}\\left(x+y+z\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkVvc1gV0n/3vctIfdCga7BWAV6Hot8PZpHVZcSNnzfwTomiybi+UcYLpl9IDb09fW3GAbya69FFUWYc3u5k63a8II003/3BXxjyiM8pLII8RskcLwm+2BGpNoxeLb6kaibgYg8/1iDoZts7IrJa3FQZPPVCWHJslu8Aaaa8flnHnql8XXPq6bNQlMm+36iNhmvvtFFwhWg0YwYRghPBBH5pEPyZ7DOOfo++3arnOi9/A==" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x+y+z\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x+y+z\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(y\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(z\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmuHQGTAre0/umYO3/E+LF4lyEB4JYjIUjkjbDZ4tfSJ+yeROYBotscHIZETI6FSe7NWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CdhSH/V18j9Kf/3yKXdVwr8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(y\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(y\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAloe9eTz2HvO8muyM4Z1kKdlyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CbAGGQy6xiZ+c/G1JYEHGagkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(z\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(z\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAn3qnIUzgb1GTknQp7LHtUHlyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CVJjfiszn29cZ4RF/xvdEEkkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=1+0+0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-8\\cdot\\:2\\left(x+y+z\\right)\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=-16\\left(x+y+z\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }