derivative of 1-cos(2x)

{ "query": { "display": "derivative of $$1-\\cos\\left(2x\\right)$$", "symbolab_question": "PRE_CALC#derivative 1-\\cos(2x)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "2\\sin(2x)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1-\\cos\\left(2x\\right)\\right)=2\\sin\\left(2x\\right)$$", "input": "\\frac{d}{dx}\\left(1-\\cos\\left(2x\\right)\\right)", "steps": [ { "type": "interim", "title": "Simplify $$1-\\cos\\left(2x\\right):{\\quad}2\\sin^{2}\\left(x\\right)$$", "input": "1-\\cos\\left(2x\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "1-\\cos\\left(2x\\right)", "result": "=2\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2x\\right)=1-2\\sin^{2}\\left(x\\right)$$", "result": "=1-\\left(1-2\\sin^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$1-\\left(1-2\\sin^{2}\\left(x\\right)\\right):{\\quad}2\\sin^{2}\\left(x\\right)$$", "input": "1-\\left(1-2\\sin^{2}\\left(x\\right)\\right)", "result": "=2\\sin^{2}\\left(x\\right)", "steps": [ { "type": "interim", "title": "$$-\\left(1-2\\sin^{2}\\left(x\\right)\\right):{\\quad}-1+2\\sin^{2}\\left(x\\right)$$", "input": "-\\left(1-2\\sin^{2}\\left(x\\right)\\right)", "result": "=1-1+2\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(1\\right)-\\left(-2\\sin^{2}\\left(x\\right)\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-1+2\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "$$1-1=0$$", "result": "=2\\sin^{2}\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bg9NpXnBBkAGBijai46r2bw1HVk6YK+QFQ5TOVDfE9dwkKGJWEPFPk38sdJMsyPIb/CPfQAW8ovubJ3T+srxfGRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9JCn16ZkNbhjoF1SNhENjOh05cwAgoqocXLlp7R9JpYU=" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PUOM60ShVE7f9cgKGM2tAeWGPr5YQ8Jy6jHFPaQOtY+AYUxCRRMw2cScUTF07kq8MvMnPiuRieO/X9beCVqRKbWzVFA6VxLY8LrAw/d4eKt2K55v4C1F0kdb2pFpR+T8dQuyXAGa92yGFgoPrmv+IYg/ZMf1UuaiLMFDB1RdxgQzCWZC/+o+Y7VVDiCUgTiqpcWJ1sSU7tcfz48+IQx+0A==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{d}{dx}\\left(2\\sin^{2}\\left(x\\right)\\right)" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(\\sin^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}2\\sin\\left(x\\right)\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin^{2}\\left(x\\right)\\right)", "result": "=2\\sin\\left(x\\right)\\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": [ "$$f=u^{2},\\:\\:u=\\sin\\left(x\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\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": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "=2\\sin\\left(x\\right)\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqmW0Oj/QNMY5AyjzWIvgQEXi79ycACn3XhxThCpcRNo1NpEj4yUFTERoeqJRRLYHBPiZ+52xB2X1cQ6EdG5IQMzaK+6qzlLDJ0SmKPm0ac5VDf2dWnZHcUYMUvHrDi3ZzbDeSQJi3LrHXuQoUSmasbWwPs1+Gw97t4MeuaNjSYTyjkVra0ajChSguMMf9fGqJUWlZ+bnpTem3dqAxhtNKM=" } }, { "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\\:2\\sin\\left(x\\right)\\cos\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:2\\sin\\left(x\\right)\\cos\\left(x\\right):{\\quad}2\\sin\\left(2x\\right)$$", "input": "2\\cdot\\:2\\sin\\left(x\\right)\\cos\\left(x\\right)", "result": "=2\\sin\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4\\sin\\left(x\\right)\\cos\\left(x\\right)" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "4\\cos\\left(x\\right)\\sin\\left(x\\right)", "result": "=2\\sin\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$2\\sin\\left(x\\right)\\cos\\left(x\\right)=\\sin\\left(2x\\right)$$", "secondary": [ "$$\\sin\\left(x\\right)\\cos\\left(x\\right)=\\frac{\\sin\\left(2x\\right)}{2}$$" ], "result": "=4\\cdot\\:\\frac{\\sin\\left(2x\\right)}{2}" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:\\frac{\\sin\\left(2x\\right)}{2}:{\\quad}2\\sin\\left(2x\\right)$$", "input": "4\\cdot\\:\\frac{\\sin\\left(2x\\right)}{2}", "result": "=2\\sin\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sin\\left(2x\\right)\\cdot\\:4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2\\sin\\left(2x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV85qLZj+Z+0sMNMcb0ZjmicBV00rpv8+ZC6TM10tVCSHsVNgZkxWKciGvDSUO6+3MRhVHyEvbNIT0SR79tx/NrMyLGmNnLPWGf9PH3lpmjoJI7oCEZuiGyY/WdF0yDmbvqDH+R7JFitQCX1DZP4xWH/FurCc0v6+SezXpJu+Sddf4" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": 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