d/(dt)(1/6 t^3)

{ "query": { "display": "$$\\frac{d}{dt}\\left(\\frac{1}{6}t^{3}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dt}(\\frac{1}{6}t^{3})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{t^{2}}{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dt}\\left(\\frac{1}{6}t^{3}\\right)=\\frac{t^{2}}{2}$$", "input": "\\frac{d}{dt}\\left(\\frac{1}{6}t^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\frac{1}{6}\\frac{d}{dt}\\left(t^{3}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{6}\\cdot\\:3t^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{6}\\cdot\\:3t^{3-1}:{\\quad}\\frac{t^{2}}{2}$$", "input": "\\frac{1}{6}\\cdot\\:3t^{3-1}", "result": "=\\frac{t^{2}}{2}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$3-1=2$$", "result": "=3\\cdot\\:\\frac{1}{6}t^{2}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3t^{2}}{6}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3t^{2}}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{t^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZTPdCvP35NKbVf5YXx+lqy/VwMe2tk2ruLtBf1kFBenehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpi8NCr75gPJczklWenKdIy/wt9LEn7QCBUukJKctfSJKm5KWrv5y7TlmVxX/22cMlyvP7yWen9fPxvFUcgZocETdVS8hJbw9Q2IsWTtZVJ47" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "y=\\frac{t^{2}}{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }