derivative of f(x)=sin(2x^3)

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\sin\\left(2x^{3}\\right)$$", "symbolab_question": "PRE_CALC#derivative f(x)=\\sin(2x^{3})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\cos(2x^{3})\\cdot 6x^{2}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(2x^{3}\\right)\\right)=\\cos\\left(2x^{3}\\right)\\cdot\\:6x^{2}$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(2x^{3}\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\cos\\left(2x^{3}\\right)\\frac{d}{dx}\\left(2x^{3}\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(2x^{3}\\right)\\right)", "result": "=\\cos\\left(2x^{3}\\right)\\frac{d}{dx}\\left(2x^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sin\\left(u\\right),\\:\\:u=2x^{3}$$" ], "result": "=\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)\\frac{d}{dx}\\left(2x^{3}\\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(2x^{3}\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x^{3}$$", "result": "=\\cos\\left(2x^{3}\\right)\\frac{d}{dx}\\left(2x^{3}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYh6yUCRk/MnA93xfZq2OhuRA3FfKPmGy5Qg8krE7Rij3OiaLJuL5RxgumX0gNvT19bcYBvJrr0UVRZhze7mTrdpl7Ws2NPXTEEtQb41WJp+aZZJt1TSoYv496p+Uul8xg9TDdx49aXlLhyuJXI0khpnvbBmbuQNTF0TphKZ8RuvabkCo9FlyJUsOgE5aAtC5rBG0MJHrQfXzSB22kHRiQ+E=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x^{3}\\right)=6x^{2}$$", "input": "\\frac{d}{dx}\\left(2x^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(x^{3}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2\\cdot\\:3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=6x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnw/L9p+IMyouNhbzU7bNo6TdaV09PMxEKZ9FieghTFwWn+xgzF413/kfGYgWrraBxHO0oTnnZveyzJ4AtC1ZGNjDT5Dj/fM73/u0bafjbUvQ9geJFbKg/ol2YLcmIH9YSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\cos\\left(2x^{3}\\right)\\cdot\\:6x^{2}" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\cos(2x^{3})\\cdot 6x^{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }