derivative of 8cos^4(x)

{ "query": { "display": "$$\\frac{d}{dx}\\left(8\\cos^{4}\\left(x\\right)\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dx}(8\\cos^{4}(x))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-32\\cos^{3}(x)\\sin(x)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(8\\cos^{4}\\left(x\\right)\\right)=-32\\cos^{3}\\left(x\\right)\\sin\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(8\\cos^{4}\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=8\\frac{d}{dx}\\left(\\cos^{4}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Apply the chain rule:$${\\quad}4\\left(\\cos\\left(x\\right)\\right)^{3}\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(\\cos^{4}\\left(x\\right)\\right)", "result": "=4\\left(\\cos\\left(x\\right)\\right)^{3}\\frac{d}{dx}\\left(\\cos\\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": [ "$$f=u^{4},\\:\\:u=\\cos\\left(x\\right)$$" ], "result": "=\\frac{d}{du}\\left(u^{4}\\right)\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{4}\\right)=4u^{3}$$", "input": "\\frac{d}{du}\\left(u^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4u^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4u^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYl3oEe4EC6n4B8Xgz0cKnkqk3hxk9aCfAWodBRxXgUexuzo0Fl6VhneutEteidn+Kv8//6/nV5O4fb8Xgwi7mapNFxUvwBeni+JEIFAdbegzYEOcQgyW4n51cR7AeMAm7Q==" } }, { "type": "step", "result": "=4u^{3}\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "=4\\left(\\cos\\left(x\\right)\\right)^{3}\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrJGzPoc4pYL6rW4BKj64u8Xi79ycACn3XhxThCpcRNo1NpEj4yUFTERoeqJRRLYHBPiZ+52xB2X1cQ6EdG5IQNDbz+rnjB7niilcMPHiaK/XAEoyFJ/6kB4slm6YrOPDgYZgssjn1g+XYlYIcWhvw5kS3dlcCKpQTQcheuut7MkqkSK49J17AMxV6yOujf/eqTH+HXrxKfWuw8Vmw3VWoA=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$", "result": "=-\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoTIPsH/5VFEfonU6bvi80j8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zA92cpyjnQxeYFWLLJRXAqymcxh5GfxfsNed5mphvPA8hRo8/W0Hq8WASQDG4Qt2SWMOnp3ra3sqxOu3sWEDlnw=" } }, { "type": "step", "result": "=8\\cdot\\:4\\left(\\cos\\left(x\\right)\\right)^{3}\\left(-\\sin\\left(x\\right)\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-32\\cos^{3}\\left(x\\right)\\sin\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-32\\cos^{3}(x)\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }