derivative of f(x)= 1/16 x^4+1/(2x^2)

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\frac{1}{16}x^{4}+\\frac{1}{2x^{2}}$$", "symbolab_question": "PRE_CALC#derivative f(x)=\\frac{1}{16}x^{4}+\\frac{1}{2x^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{x^{3}}{4}-\\frac{1}{x^{3}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{1}{16}x^{4}+\\frac{1}{2x^{2}}\\right)=\\frac{x^{3}}{4}-\\frac{1}{x^{3}}$$", "input": "\\frac{d}{dx}\\left(\\frac{1}{16}x^{4}+\\frac{1}{2x^{2}}\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{1}{16}x^{4}+\\frac{1}{2x^{2}}:{\\quad}\\frac{x^{4}}{16}+\\frac{1}{2x^{2}}$$", "input": "\\frac{1}{16}x^{4}+\\frac{1}{2x^{2}}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{16}x^{4}=\\frac{x^{4}}{16}$$", "input": "\\frac{1}{16}x^{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x^{4}}{16}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{4}=x^{4}$$", "result": "=\\frac{x^{4}}{16}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79rgdyfrE8kg+mZG8x+qgbbs17PPbVicYM1JwSbyTGc3MwViaLUXkeD+JukROhWdjvKVbIIQY8KAFGCayZN9C5KvKzRUWvR7H7TI+qhAXIp0SMSeoGlUm9mXvoOnEIrKxhpuG/oExc1hEUzgjAIr3z8/HI8xbZvLcAnAW0RanvGo=" } }, { "type": "step", "result": "=\\frac{x^{4}}{16}+\\frac{1}{2x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(\\frac{x^{4}}{16}+\\frac{1}{2x^{2}}\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(\\frac{x^{4}}{16}\\right)+\\frac{d}{dx}\\left(\\frac{1}{2x^{2}}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{x^{4}}{16}\\right)=\\frac{x^{3}}{4}$$", "input": "\\frac{d}{dx}\\left(\\frac{x^{4}}{16}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\frac{1}{16}\\frac{d}{dx}\\left(x^{4}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{16}\\cdot\\:4x^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{16}\\cdot\\:4x^{4-1}:{\\quad}\\frac{x^{3}}{4}$$", "input": "\\frac{1}{16}\\cdot\\:4x^{4-1}", "result": "=\\frac{x^{3}}{4}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$4-1=3$$", "result": "=4\\cdot\\:\\frac{1}{16}x^{3}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4x^{3}}{16}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=\\frac{4x^{3}}{16}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\frac{x^{3}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7klyBl7FLqtGgC44uj6hTeQv+H2Zr7pVedn9f/oouXfxV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAllw1lL1m4GtyxyX4CeY8W4WBBTEk/JQ2cZ9WKuRzClU7QG+dQXHhPmuaaYlTGyzk2RSyMuC9dru4d00/FmWMpp2gqRbnXXx1V9UugVBSWOG9" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{1}{2x^{2}}\\right)=-\\frac{1}{x^{3}}$$", "input": "\\frac{d}{dx}\\left(\\frac{1}{2x^{2}}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\frac{1}{2}\\frac{d}{dx}\\left(\\frac{1}{x^{2}}\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=\\frac{1}{2}\\frac{d}{dx}\\left(x^{-2}\\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": "=\\frac{1}{2}\\left(-2x^{-2-1}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\left(-2x^{-2-1}\\right):{\\quad}-\\frac{1}{x^{3}}$$", "input": "\\frac{1}{2}\\left(-2x^{-2-1}\\right)", "result": "=-\\frac{1}{x^{3}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2}\\cdot\\:2x^{-2-1}" }, { "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": "step", "primary": "Subtract the numbers: $$-2-1=-3$$", "result": "=-1\\cdot\\:x^{-3}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-3}=\\frac{1}{x^{3}}$$" ], "result": "=-1\\cdot\\:\\frac{1}{x^{3}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{3}}=\\frac{1}{x^{3}}$$", "result": "=-\\frac{1}{x^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79jHpDL2BX3rbBpw0SEa3igPYTnmBQJBxsPQVut3sBfvNGoPE9TME3q+OPmgkv2RQrRKMQ0pA8A67jlmbMMkBF2rX9hHKOj4O495tZABVEFR6pfF1z6umzUJTJvt+ojYZY4LVEbDWfJhdqtpUA6/Oh24igFqOPVbNLztNysdaJ5Utg8J8fpq6qOKZMZslXySI" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{x^{3}}{4}-\\frac{1}{x^{3}}" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{x^{3}}{4}-\\frac{1}{x^{3}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }