derivative of-1cos(t)

{ "query": { "display": "derivative of $$-1\\cos\\left(t\\right)$$", "symbolab_question": "PRE_CALC#derivative -1\\cos(t)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\sin(t)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dt}\\left(-1\\cdot\\:\\cos\\left(t\\right)\\right)=\\sin\\left(t\\right)$$", "input": "\\frac{d}{dt}\\left(-1\\cdot\\:\\cos\\left(t\\right)\\right)", "steps": [ { "type": "interim", "title": "Simplify $$-1\\cdot\\:\\cos\\left(t\\right):{\\quad}-\\cos\\left(t\\right)$$", "input": "-1\\cdot\\:\\cos\\left(t\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos\\left(t\\right)=\\cos\\left(t\\right)$$", "result": "=-\\cos\\left(t\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{d}{dt}\\left(-\\cos\\left(t\\right)\\right)" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{d}{dt}\\left(\\cos\\left(t\\right)\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dt}\\left(\\cos\\left(t\\right)\\right)=-\\sin\\left(t\\right)$$", "result": "=-\\left(-\\sin\\left(t\\right)\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\sin\\left(t\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "y=\\sin(t)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }