derivative of f(x)=sin(3x)

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\sin\\left(3x\\right)$$", "symbolab_question": "PRE_CALC#derivative f(x)=\\sin(3x)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\cos(3x)\\cdot 3", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(3x\\right)\\right)=\\cos\\left(3x\\right)\\cdot\\:3$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(3x\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\cos\\left(3x\\right)\\frac{d}{dx}\\left(3x\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(3x\\right)\\right)", "result": "=\\cos\\left(3x\\right)\\frac{d}{dx}\\left(3x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sin\\left(u\\right),\\:\\:u=3x$$" ], "result": "=\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)\\frac{d}{dx}\\left(3x\\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(3x\\right)" }, { "type": "step", "primary": "Substitute back $$u=3x$$", "result": "=\\cos\\left(3x\\right)\\frac{d}{dx}\\left(3x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqob0EQWSyJhvgWcUxC3Q2mQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEyA4M871BpNnK+ikVC1jucRBlkWsvrUWlvTVahbtSUEJCGk8vIJisuT2N3pfkW1JpbGddanbms4cigCpjwsLZ/8wk+QLk42sL8qUHm7drm4ACS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x\\right)=3$$", "input": "\\frac{d}{dx}\\left(3x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsUnAVaDXCLQOuynYB+k3fTZGku9zFkxwe1dTH8vycb9BbqPJ4kl+ElAajU+EBTcW1NbbqpyK7JQEZdATEJR51i3CwF9WgU8/+rFW242YnYD" } }, { "type": "step", "result": "=\\cos\\left(3x\\right)\\cdot\\:3" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\cos(3x)\\cdot 3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }