derivative of f(x)=(2sin^3(x)-5x)^{5/3}

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{5}{3}}$$", "symbolab_question": "PRE_CALC#derivative f(x)=(2\\sin^{3}(x)-5x)^{\\frac{5}{3}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{5(6\\sin^{2}(x)\\cos(x)-5)(2\\sin^{3}(x)-5x)^{\\frac{2}{3}}}{3}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{5}{3}}\\right)=\\frac{5\\left(6\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)-5\\right)\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{2}{3}}}{3}$$", "input": "\\frac{d}{dx}\\left(\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{5}{3}}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{5\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{2}{3}}}{3}\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)-5x\\right)$$", "input": "\\frac{d}{dx}\\left(\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{5}{3}}\\right)", "result": "=\\frac{5\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{2}{3}}}{3}\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)-5x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{\\frac{5}{3}},\\:\\:u=\\left(2\\sin^{3}\\left(x\\right)-5x\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{\\frac{5}{3}}\\right)\\frac{d}{dx}\\left(\\left(2\\sin^{3}\\left(x\\right)-5x\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{\\frac{5}{3}}\\right)=\\frac{5u^{\\frac{2}{3}}}{3}$$", "input": "\\frac{d}{du}\\left(u^{\\frac{5}{3}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{5}{3}u^{\\frac{5}{3}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{5}{3}u^{\\frac{5}{3}-1}:{\\quad}\\frac{5u^{\\frac{2}{3}}}{3}$$", "input": "\\frac{5}{3}u^{\\frac{5}{3}-1}", "result": "=\\frac{5u^{\\frac{2}{3}}}{3}", "steps": [ { "type": "interim", "title": "$$u^{\\frac{5}{3}-1}=u^{\\frac{2}{3}}$$", "input": "u^{\\frac{5}{3}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{5}{3}-1:{\\quad}\\frac{2}{3}$$", "input": "\\frac{5}{3}-1", "result": "=u^{\\frac{2}{3}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=-\\frac{1\\cdot\\:3}{3}+\\frac{5}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:3+5}{3}" }, { "type": "interim", "title": "$$-1\\cdot\\:3+5=2$$", "input": "-1\\cdot\\:3+5", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=-3+5" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-3+5=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NKzIdLoxfOWctINKoYkNv1XTSum/z5kLpMzXS1UJIezBOIcT9LwoPFpLgX7k/D9TSuvnmHHmwzaA2REu2aAnP84ka8F9WsyASLd6olVazQQ=" } }, { "type": "step", "result": "=\\frac{2}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pdTyORg72Xwp/fTU7gw1m+0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjrAnm+0dWV2G5ts+qDasw1P8//6/nV5O4fb8Xgwi7marpZlpPA1sHTvP5viOtpjQ/MMgFYo0k1nCcObJSAQu6gs67uJicgMpLzW5Yu+HQ/jQ=" } }, { "type": "step", "result": "=\\frac{5}{3}u^{\\frac{2}{3}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{5u^{\\frac{2}{3}}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HEcdymjQozFmRk5Oh5FBzbCttvJlYc8bs4y+5RPcRqgAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJgwOKsVAiPOfX+lJC9WiIMvDakV5xtzrC6sUYTlsYKLn1sD7NfhsPe7eDHrmjY0mE8DwQcN28geUT7aUsumC7doBDKD2XvEYcwnHxnLYU/iA2sn2RC+c/BDkqFyxK4HceQ==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{5u^{\\frac{2}{3}}}{3}\\frac{d}{dx}\\left(\\left(2\\sin^{3}\\left(x\\right)-5x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\left(2\\sin^{3}\\left(x\\right)-5x\\right)$$", "result": "=\\frac{5\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{2}{3}}}{3}\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)-5x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsKTtXGfF/MkSl78WXXxyvUjNNt59C2HWoB/JNL4KKqbC7vGbZ2u0RQZ8l5TbOwGxgeNnQmZ4ALdlISYzFZE1i7peVGypUVICZb3mm1LEudSIBz+GKlpIeTjh2kq0P5fEfH0HO6o5Ykj5thooWcS37OimxKlJxew+85S9cHc2ICABh7Z+NaTHtThPA34waXeGfnVNlUlUrdnxP5m4O9D1TyD63WlyOmbnHNz6QflRNyR8LfSxJ+0AgVLpCSnLX0iSs2zS5GGbCZa1X3a2T5z/xm7+YNMdwDOICJh407OOWgm" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)-5x\\right)=6\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)-5$$", "input": "\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)-5x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)\\right)-\\frac{d}{dx}\\left(5x\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)\\right)=6\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(2\\sin^{3}\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(\\sin^{3}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}3\\left(\\sin\\left(x\\right)\\right)^{2}\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin^{3}\\left(x\\right)\\right)", "result": "=3\\left(\\sin\\left(x\\right)\\right)^{2}\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{3},\\:\\:u=\\sin\\left(x\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{3}\\right)\\frac{d}{dx}\\left(\\sin\\left(x\\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(\\sin\\left(x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "=3\\left(\\sin\\left(x\\right)\\right)^{2}\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqNxjWGE58H9oI7CDLL2A/4Xi79ycACn3XhxThCpcRNo1NpEj4yUFTERoeqJRRLYHBPiZ+52xB2X1cQ6EdG5IQO8Lf1Cr1RIMG05OFJAphwe/anAJHqrRTzOIpfDjrc9tHVYsbkCD4DRWIECF2dTStlkS3dlcCKpQTQcheuut7MkqkSK49J17AMxV6yOujf/eqTH+HXrxKfWuw8Vmw3VWoA=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)=\\cos\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)=\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgOt2FhQQwx0GxLGzv2mPOv8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaugB66mSUqneplfTkjggryzA+YUX37Aa/AAEf1Hkty8FUj7LPbFLewMJWlj8VtjhXr5J/4xg9Nn6C/zrAXreziPc=" } }, { "type": "step", "result": "=2\\cdot\\:3\\left(\\sin\\left(x\\right)\\right)^{2}\\cos\\left(x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=6\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5x\\right)=5$$", "input": "\\frac{d}{dx}\\left(5x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=5\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=5\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=5", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmZ9X+BZ/JyYYY1hJF6v/r/ZGku9zFkxwe1dTH8vycb9dZwuGwx+eQcEBv+dY3CjTFNbbqpyK7JQEZdATEJR51hJPYdcyzxncaY/pkYJ4XZq" } }, { "type": "step", "result": "=6\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)-5" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{5\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{2}{3}}}{3}\\left(6\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)-5\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{5\\left(6\\sin^{2}\\left(x\\right)\\cos\\left(x\\right)-5\\right)\\left(2\\sin^{3}\\left(x\\right)-5x\\right)^{\\frac{2}{3}}}{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{5(6\\sin^{2}(x)\\cos(x)-5)(2\\sin^{3}(x)-5x)^{\\frac{2}{3}}}{3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }