derivative of cot^3(4x+1)

{ "query": { "display": "derivative of $$\\cot^{3}\\left(4x+1\\right)$$", "symbolab_question": "PRE_CALC#derivative \\cot^{3}(4x+1)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-12\\cot^{2}(4x+1)\\csc^{2}(4x+1)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\cot^{3}\\left(4x+1\\right)\\right)=-12\\cot^{2}\\left(4x+1\\right)\\csc^{2}\\left(4x+1\\right)$$", "input": "\\frac{d}{dx}\\left(\\cot^{3}\\left(4x+1\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(\\cot\\left(4x+1\\right)\\right)^{2}\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(\\cot^{3}\\left(4x+1\\right)\\right)", "result": "=3\\left(\\cot\\left(4x+1\\right)\\right)^{2}\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\cot\\left(4x+1\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{3}\\right)=3u^{2}$$", "input": "\\frac{d}{du}\\left(u^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3u^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3u^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjNrXZy15fg+DDm4IZ/kmJKk3hxk9aCfAWodBRxXgUexYrhQEWwJJMG//0JI2CMZRv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegz8nAGF3XMCO2i5cp0uYL6iA==" } }, { "type": "step", "result": "=3u^{2}\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\cot\\left(4x+1\\right)$$", "result": "=3\\left(\\cot\\left(4x+1\\right)\\right)^{2}\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYj43n+FXaeUFGGvDGbsQA0fGOOatBGQTlDG0wQgpIs2iZ3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJVZ0IlZjWUBPpgqSqITgp3IiZDXPhAh+hCLG5hLyCtdamMVewE1NsVL9/sbYcmjxKXirT5kTRDKMp+g+xI91tLdeqXxdc+rps1CUyb7fqI2Ga++0UXCFaDRjBhGCE8EEfmkQ/JnsM45+j77dquc6L38" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)=-\\csc^{2}\\left(4x+1\\right)\\cdot\\:4$$", "input": "\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}-\\csc^{2}\\left(4x+1\\right)\\frac{d}{dx}\\left(4x+1\\right)$$", "input": "\\frac{d}{dx}\\left(\\cot\\left(4x+1\\right)\\right)", "result": "=-\\csc^{2}\\left(4x+1\\right)\\frac{d}{dx}\\left(4x+1\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\cot\\left(u\\right),\\:\\:u=4x+1$$" ], "result": "=\\frac{d}{du}\\left(\\cot\\left(u\\right)\\right)\\frac{d}{dx}\\left(4x+1\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\cot\\left(u\\right)\\right)=-\\csc^{2}\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\cot\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\cot\\left(u\\right)\\right)=-\\csc^{2}\\left(u\\right)$$", "result": "=-\\csc^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnewLus0Iwt2GIfaj7BCRBb8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zHhPvW42uzNHy1Jo8GdX3ZkOG38IleojCyebAtZy+3Tmd0nSH53xBDFBL5CvTmy1nMwWHuopLT4qX7pNiKxZf4GwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=-\\csc^{2}\\left(u\\right)\\frac{d}{dx}\\left(4x+1\\right)" }, { "type": "step", "primary": "Substitute back $$u=4x+1$$", "result": "=-\\csc^{2}\\left(4x+1\\right)\\frac{d}{dx}\\left(4x+1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpu5TUg2reqEW1PPINyab4a6FuYqVNTsV6Yi7DZx8hD8dLl7DeVd7l7l/uUT/v1GhNJLZOXSTCXkMFKW90A2Pi//PSprZJb6eDDMxycYrxcFlLUoBlq475LabrK+6JyyyWwlP9qAv+fn0mBcAfOQAeeBBTEk/JQ2cZ9WKuRzClU7Gzcoi/GdadAN9Y/6IEMNsKuwb2V9CUO2vaxv8HHl0nU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x+1\\right)=4$$", "input": "\\frac{d}{dx}\\left(4x+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(4x\\right)+\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x\\right)=4$$", "input": "\\frac{d}{dx}\\left(4x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=4\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=4", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjTQT54ZonDS7voDaGrIxyrZGku9zFkxwe1dTH8vycb9lXydCAnPpeOlbdfKkqvrL1NbbqpyK7JQEZdATEJR51jxWDAE4KteS3Xb/OKFHmJV" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=4+0" }, { "type": "step", "primary": "Simplify", "result": "=4", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\csc^{2}\\left(4x+1\\right)\\cdot\\:4" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=3\\left(\\cot\\left(4x+1\\right)\\right)^{2}\\left(-\\csc^{2}\\left(4x+1\\right)\\cdot\\:4\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-12\\cot^{2}\\left(4x+1\\right)\\csc^{2}\\left(4x+1\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-12\\cot^{2}(4x+1)\\csc^{2}(4x+1)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }