derivative of 9(sin(x)+xcos(x))

{ "query": { "display": "derivative of $$9\\left(\\sin\\left(x\\right)+x\\cos\\left(x\\right)\\right)$$", "symbolab_question": "PRE_CALC#derivative 9(\\sin(x)+x\\cos(x))" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "9(2\\cos(x)-x\\sin(x))", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(9\\left(\\sin\\left(x\\right)+x\\cos\\left(x\\right)\\right)\\right)=9\\left(2\\cos\\left(x\\right)-x\\sin\\left(x\\right)\\right)$$", "input": "\\frac{d}{dx}\\left(9\\left(\\sin\\left(x\\right)+x\\cos\\left(x\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=9\\frac{d}{dx}\\left(\\sin\\left(x\\right)+x\\cos\\left(x\\right)\\right)" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=9\\left(\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)+\\frac{d}{dx}\\left(x\\cos\\left(x\\right)\\right)\\right)" }, { "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": "interim", "title": "$$\\frac{d}{dx}\\left(x\\cos\\left(x\\right)\\right)=\\cos\\left(x\\right)-x\\sin\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(x\\cos\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x,\\:g=\\cos\\left(x\\right)$$" ], "result": "=\\frac{dx}{dx}\\cos\\left(x\\right)+\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)x", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "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": "=1\\cdot\\:\\cos\\left(x\\right)+\\left(-\\sin\\left(x\\right)\\right)x" }, { "type": "step", "primary": "Simplify", "result": "=\\cos\\left(x\\right)-x\\sin\\left(x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=9\\left(\\cos\\left(x\\right)+\\cos\\left(x\\right)-x\\sin\\left(x\\right)\\right)" }, { "type": "step", "primary": "Simplify", "result": "=9\\left(2\\cos\\left(x\\right)-x\\sin\\left(x\\right)\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=9(2\\cos(x)-x\\sin(x))" }, "showViewLarger": true } }, "meta": { "showVerify": true } }