d/(dt)(tan(e^{2t})+e^{tan(2t)})

{ "query": { "display": "$$\\frac{d}{dt}\\left(\\tan\\left(e^{2t}\\right)+e^{\\tan\\left(2t\\right)}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dt}(\\tan(e^{2t})+e^{\\tan(2t)})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\sec^{2}(e^{2t})e^{2t}\\cdot 2+e^{\\tan(2t)}\\sec^{2}(2t)\\cdot 2", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dt}\\left(\\tan\\left(e^{2t}\\right)+e^{\\tan\\left(2t\\right)}\\right)=\\sec^{2}\\left(e^{2t}\\right)e^{2t}\\cdot\\:2+e^{\\tan\\left(2t\\right)}\\sec^{2}\\left(2t\\right)\\cdot\\:2$$", "input": "\\frac{d}{dt}\\left(\\tan\\left(e^{2t}\\right)+e^{\\tan\\left(2t\\right)}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dt}\\left(\\tan\\left(e^{2t}\\right)\\right)+\\frac{d}{dt}\\left(e^{\\tan\\left(2t\\right)}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(\\tan\\left(e^{2t}\\right)\\right)=\\sec^{2}\\left(e^{2t}\\right)e^{2t}\\cdot\\:2$$", "input": "\\frac{d}{dt}\\left(\\tan\\left(e^{2t}\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\sec^{2}\\left(e^{2t}\\right)\\frac{d}{dt}\\left(e^{2t}\\right)$$", "input": "\\frac{d}{dt}\\left(\\tan\\left(e^{2t}\\right)\\right)", "result": "=\\sec^{2}\\left(e^{2t}\\right)\\frac{d}{dt}\\left(e^{2t}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\tan\\left(u\\right),\\:\\:u=e^{2t}$$" ], "result": "=\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)\\frac{d}{dt}\\left(e^{2t}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)=\\sec^{2}\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)=\\sec^{2}\\left(u\\right)$$", "result": "=\\sec^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmL1ED/uAUxad8tgBjyUCcz8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaupzyo2aG+HffnS5jYi+BBaLNWyGcX6HZt1LGXH2QGa+LYvUQP+4BTFp3y2AGPJQJzK+EVs14Vj10NBJLj8DnhDgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\sec^{2}\\left(u\\right)\\frac{d}{dt}\\left(e^{2t}\\right)" }, { "type": "step", "primary": "Substitute back $$u=e^{2t}$$", "result": "=\\sec^{2}\\left(e^{2t}\\right)\\frac{d}{dt}\\left(e^{2t}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrcTDNRvcvEho8HDbV4drVk/vZjDpdOE0AID43d7PGn4OiaLJuL5RxgumX0gNvT19bcYBvJrr0UVRZhze7mTrdqS1OYZTh+dm0knDKJbDsgMH1/Y7zGYaCaWyEd1YoErs82a7bsd04+AldyVMy3FC+whpPLyCYrLk9jd6X5FtSaWxnXWp25rOHIoAqY8LC2f/MJPkC5ONrC/KlB5u3a5uAAkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(e^{2t}\\right)=e^{2t}\\cdot\\:2$$", "input": "\\frac{d}{dt}\\left(e^{2t}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{2t}\\frac{d}{dt}\\left(2t\\right)$$", "input": "\\frac{d}{dt}\\left(e^{2t}\\right)", "result": "=e^{2t}\\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": [ "$$f=e^{u},\\:\\:u=2t$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dt}\\left(2t\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "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(2t\\right)" }, { "type": "step", "primary": "Substitute back $$u=2t$$", "result": "=e^{2t}\\frac{d}{dt}\\left(2t\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYvMurBLeWDiX/Qq9BJrypZGTdaV09PMxEKZ9FieghTFwc6p4sNpoW3XPzo2W3Ux6aosjLe8tD9HbrkG8vq6q9jgtQvpXWTAYqlAu+XsvwISIgyDbRTdD1JDbiV+XM4e7wvC30sSftAIFS6Qkpy19IkrNs0uRhmwmWtV92tk+c/8Zu/mDTHcAziAiYeNOzjloJg==" } }, { "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": "=e^{2t}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\sec^{2}\\left(e^{2t}\\right)e^{2t}\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(e^{\\tan\\left(2t\\right)}\\right)=e^{\\tan\\left(2t\\right)}\\sec^{2}\\left(2t\\right)\\cdot\\:2$$", "input": "\\frac{d}{dt}\\left(e^{\\tan\\left(2t\\right)}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{\\tan\\left(2t\\right)}\\frac{d}{dt}\\left(\\tan\\left(2t\\right)\\right)$$", "input": "\\frac{d}{dt}\\left(e^{\\tan\\left(2t\\right)}\\right)", "result": "=e^{\\tan\\left(2t\\right)}\\frac{d}{dt}\\left(\\tan\\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": [ "$$f=e^{u},\\:\\:u=\\tan\\left(2t\\right)$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dt}\\left(\\tan\\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(\\tan\\left(2t\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(2t\\right)$$", "result": "=e^{\\tan\\left(2t\\right)}\\frac{d}{dt}\\left(\\tan\\left(2t\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYk77s/DK3pDgxQ6hJ8yDbPKyU3/CSY1icdtlIFMUFaXyOiaLJuL5RxgumX0gNvT19bcYBvJrr0UVRZhze7mTrdqXry89vTeSk1KE/WkMdYByvsdQ+aOJfNXN2sijDMkn7e9r4g4Ws3286Y74PSv0RiBkS3dlcCKpQTQcheuut7MkqkSK49J17AMxV6yOujf/eqTH+HXrxKfWuw8Vmw3VWoA=" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(\\tan\\left(2t\\right)\\right)=\\sec^{2}\\left(2t\\right)\\cdot\\:2$$", "input": "\\frac{d}{dt}\\left(\\tan\\left(2t\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\sec^{2}\\left(2t\\right)\\frac{d}{dt}\\left(2t\\right)$$", "input": "\\frac{d}{dt}\\left(\\tan\\left(2t\\right)\\right)", "result": "=\\sec^{2}\\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": [ "$$f=\\tan\\left(u\\right),\\:\\:u=2t$$" ], "result": "=\\frac{d}{du}\\left(\\tan\\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": "$$\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)=\\sec^{2}\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)=\\sec^{2}\\left(u\\right)$$", "result": "=\\sec^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmL1ED/uAUxad8tgBjyUCcz8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaupzyo2aG+HffnS5jYi+BBaLNWyGcX6HZt1LGXH2QGa+LYvUQP+4BTFp3y2AGPJQJzK+EVs14Vj10NBJLj8DnhDgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\sec^{2}\\left(u\\right)\\frac{d}{dt}\\left(2t\\right)" }, { "type": "step", "primary": "Substitute back $$u=2t$$", "result": "=\\sec^{2}\\left(2t\\right)\\frac{d}{dt}\\left(2t\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtkhu6tEsPIjFsZ13d2rAiaQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMExWJrXN7Mf59jefKycgLxn6kq62Dy8rkzHz4UH6eMOllE8TJMEoIyL+Fjt49yia4wVK28PwYoM0kk3yqK7RdFxVkBgC9tEqqRFUNoMEeRIG0PBgeR+YcdPN7cleTcuOImk=" } }, { "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": "=\\sec^{2}\\left(2t\\right)\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{\\tan\\left(2t\\right)}\\sec^{2}\\left(2t\\right)\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\sec^{2}\\left(e^{2t}\\right)e^{2t}\\cdot\\:2+e^{\\tan\\left(2t\\right)}\\sec^{2}\\left(2t\\right)\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "y=\\sec^{2}(e^{2t})e^{2t}\\cdot 2+e^{\\tan(2t)}\\sec^{2}(2t)\\cdot 2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }