derivative of f(t)=e^{7tsin(2t)}

{ "query": { "display": "derivative of $$f\\left(t\\right)=e^{7t\\sin\\left(2t\\right)}$$", "symbolab_question": "PRE_CALC#derivative f(t)=e^{7t\\sin(2t)}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "e^{7t\\sin(2t)}\\cdot 7(\\sin(2t)+2t\\cos(2t))", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dt}\\left(e^{7t\\sin\\left(2t\\right)}\\right)=e^{7t\\sin\\left(2t\\right)}\\cdot\\:7\\left(\\sin\\left(2t\\right)+2t\\cos\\left(2t\\right)\\right)$$", "input": "\\frac{d}{dt}\\left(e^{7t\\sin\\left(2t\\right)}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{7t\\sin\\left(2t\\right)}\\frac{d}{dt}\\left(7t\\sin\\left(2t\\right)\\right)$$", "input": "\\frac{d}{dt}\\left(e^{7t\\sin\\left(2t\\right)}\\right)", "result": "=e^{7t\\sin\\left(2t\\right)}\\frac{d}{dt}\\left(7t\\sin\\left(2t\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=7t\\sin\\left(2t\\right)$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dt}\\left(7t\\sin\\left(2t\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{d}{du}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqCr3EWRZw3L4+rHTTdVG0Ok3hxk9aCfAWodBRxXgUexwx+RE9MtjN5hKMwTI7fffj/L0MoYg+CUn6oyL3EO7YrHahlpzKGY893KZ4T4i4Tv3RCXWsqiNx7T9zOhL5sYfw==" } }, { "type": "step", "result": "=e^{u}\\frac{d}{dt}\\left(7t\\sin\\left(2t\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=7t\\sin\\left(2t\\right)$$", "result": "=e^{7t\\sin\\left(2t\\right)}\\frac{d}{dt}\\left(7t\\sin\\left(2t\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYls2ZZtBh1E0B9bIUIV5mzJKEa7fvDtfXLRy+ejwPIWXZ3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJUESw8SVUCd7jm8ydTSENwh9srLNjFnWyCQQYEaaJTmldFT1ZZrgesDnUMoH2r9AxmOnhsDtimYCHpl65JYhnlh5sNavoYxRIysyUYGC00RkiLB9NzWGwsvAagupdR73VIGdD99LdN1jlwlY2h9xQom" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(7t\\sin\\left(2t\\right)\\right)=7\\left(\\sin\\left(2t\\right)+2t\\cos\\left(2t\\right)\\right)$$", "input": "\\frac{d}{dt}\\left(7t\\sin\\left(2t\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=7\\frac{d}{dt}\\left(t\\sin\\left(2t\\right)\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=t,\\:g=\\sin\\left(2t\\right)$$" ], "result": "=7\\left(\\frac{dt}{dt}\\sin\\left(2t\\right)+\\frac{d}{dt}\\left(\\sin\\left(2t\\right)\\right)t\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{dt}{dt}=1$$", "input": "\\frac{dt}{dt}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dt}{dt}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuWV6zCUVy7FvtVpq63L1y1jqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIUVox5V37RRgiM2tHP1hZLC9fXkG27pZ636yeVofyg8V" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(\\sin\\left(2t\\right)\\right)=\\cos\\left(2t\\right)\\cdot\\:2$$", "input": "\\frac{d}{dt}\\left(\\sin\\left(2t\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\cos\\left(2t\\right)\\frac{d}{dt}\\left(2t\\right)$$", "input": "\\frac{d}{dt}\\left(\\sin\\left(2t\\right)\\right)", "result": "=\\cos\\left(2t\\right)\\frac{d}{dt}\\left(2t\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sin\\left(u\\right),\\:\\:u=2t$$" ], "result": "=\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)\\frac{d}{dt}\\left(2t\\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}{dt}\\left(2t\\right)" }, { "type": "step", "primary": "Substitute back $$u=2t$$", "result": "=\\cos\\left(2t\\right)\\frac{d}{dt}\\left(2t\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpjhm1FoSu170TSP++s+5saQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEzsHvu5sPU4JAMPICHcZGbEyEz6t6GQXbBrwZLQpKNFJCGk8vIJisuT2N3pfkW1JpbGddanbms4cigCpjwsLZ/8wk+QLk42sL8qUHm7drm4ACS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(2t\\right)=2$$", "input": "\\frac{d}{dt}\\left(2t\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dt}{dt}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dt}{dt}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYppPrQUb3hSMXzMICgYQqRHZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51iWy7jOLxIpKjsdAhzQgvX5" } }, { "type": "step", "result": "=\\cos\\left(2t\\right)\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=7\\left(1\\cdot\\:\\sin\\left(2t\\right)+\\cos\\left(2t\\right)\\cdot\\:2t\\right)" }, { "type": "step", "primary": "Simplify", "result": "=7\\left(\\sin\\left(2t\\right)+2t\\cos\\left(2t\\right)\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{7t\\sin\\left(2t\\right)}\\cdot\\:7\\left(\\sin\\left(2t\\right)+2t\\cos\\left(2t\\right)\\right)" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "y=e^{7t\\sin(2t)}\\cdot 7(\\sin(2t)+2t\\cos(2t))" }, "showViewLarger": true } }, "meta": { "showVerify": true } }