derivative of y=sin(pix)

{ "query": { "display": "derivative of $$y=\\sin\\left(πx\\right)$$", "symbolab_question": "PRE_CALC#derivative y=\\sin(πx)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\cos(πx)π", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(πx\\right)\\right)=\\cos\\left(πx\\right)π$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(πx\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\cos\\left(πx\\right)\\frac{d}{dx}\\left(πx\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(πx\\right)\\right)", "result": "=\\cos\\left(πx\\right)\\frac{d}{dx}\\left(πx\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sin\\left(u\\right),\\:\\:u=πx$$" ], "result": "=\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)\\frac{d}{dx}\\left(πx\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$", "result": "=\\cos\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgerJLn9ae0g0/tUjnRuL1v8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaugHqnJiEuQ8NpaCSOBx7rI4+YUX37Aa/AAEf1Hkty8FUcUM2sEdv7dIX0bKYOeE19OmgDIY2KBZfpU9cYqvCXz4=" } }, { "type": "step", "result": "=\\cos\\left(u\\right)\\frac{d}{dx}\\left(πx\\right)" }, { "type": "step", "primary": "Substitute back $$u=πx$$", "result": "=\\cos\\left(πx\\right)\\frac{d}{dx}\\left(πx\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYk3rfqvPk6evrjy//JsdbQixQdzAEijiIbf6fw1DG6z4urf72DA1fe1V1/79l0JjTJ1qmUmvlTu9Qd9Ei/4xn2uaUAtNGOSmAmiy3MhEPmluZVoOLKq/b8XTL4hRiGsifv+WKLY3ORDPvR7heLFLlyYEuDOVaQvKofqHoY5jNapszx1SRmkdwriKeTV96yww0ImpXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(πx\\right)=π$$", "input": "\\frac{d}{dx}\\left(πx\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=π\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=π1" }, { "type": "step", "primary": "Simplify", "result": "=π", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkvop4HGMj9OAf7XT01mx2KXIQHgliMhSOSNsNni19In0Yt4K7WU4LRO+rQwwC1KNg4bfwiV6iMLJ5sC1nL7dOb49urfXN/wJ+UmdGUWJl/gsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=\\cos\\left(πx\\right)π" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\cos(πx)π", "displayFormula": "y=\\cos(πx)π", "derivativeFormula": "-π^{2}\\sin(πx)", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "calculatePoints": true } ] }, "pointsToDraw": { "pointsLatex": [ "(-\\frac{7}{2},0)", "(-\\frac{3}{2},0)", "(\\frac{1}{2},0)", "(\\frac{5}{2},0)", "(\\frac{9}{2},0)", "(-\\frac{5}{2},0)", "(-\\frac{1}{2},0)", "(\\frac{3}{2},0)", "(\\frac{7}{2},0)", "(\\frac{11}{2},0)", "(0,π)", "(-4,π)", 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