derivative of sin(3x^2cos(2x))

{ "query": { "display": "$$\\frac{d}{dx}\\left(\\sin\\left(3\\right)x^{2}\\cos\\left(2x\\right)\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dx}(\\sin(3)x^{2}\\cos(2x))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\sin(3)(2x\\cos(2x)-2x^{2}\\sin(2x))", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(3\\right)x^{2}\\cos\\left(2x\\right)\\right)=\\sin\\left(3\\right)\\left(2x\\cos\\left(2x\\right)-2x^{2}\\sin\\left(2x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(3\\right)x^{2}\\cos\\left(2x\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\sin\\left(3\\right)\\frac{d}{dx}\\left(x^{2}\\cos\\left(2x\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x^{2},\\:g=\\cos\\left(2x\\right)$$" ], "result": "=\\sin\\left(3\\right)\\left(\\frac{d}{dx}\\left(x^{2}\\right)\\cos\\left(2x\\right)+\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)x^{2}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)=-\\sin\\left(2x\\right)\\cdot\\:2$$", "input": "\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}-\\sin\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)$$", "input": "\\frac{d}{dx}\\left(\\cos\\left(2x\\right)\\right)", "result": "=-\\sin\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\cos\\left(u\\right),\\:\\:u=2x$$" ], "result": "=\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)\\frac{d}{dx}\\left(2x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)=-\\sin\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)=-\\sin\\left(u\\right)$$", "result": "=-\\sin\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuOCaaVOQ/y0RGnrcxaPJLr8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zBTby8v9dqkicCwl97RZujimcxh5GfxfsNed5mphvPA8XbuNpg35JEo1Mz1bwBSzW9DMulhoXRrVM26A7dfZaj8=" } }, { "type": "step", "result": "=-\\sin\\left(u\\right)\\frac{d}{dx}\\left(2x\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=-\\sin\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmg4HtP7a1zZWb8Xc4KE0TqQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEybd4WCnwVDNznFKnB/ZRY2yOGqGcv9anBrVhDS9/WspXULslwBmvdshhYKD65r/iHdKZS9fj5nGO8F191IYKw7ClwHVv5rKpMbWSxPD5Uhrr8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw" } }, { "type": "step", "result": "=-\\sin\\left(2x\\right)\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\sin\\left(3\\right)\\left(2x\\cos\\left(2x\\right)+\\left(-\\sin\\left(2x\\right)\\cdot\\:2\\right)x^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\sin\\left(3\\right)\\left(2x\\cos\\left(2x\\right)-2x^{2}\\sin\\left(2x\\right)\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\sin(3)(2x\\cos(2x)-2x^{2}\\sin(2x))" }, "showViewLarger": true } }, "meta": { "showVerify": true } }