derivative of (sin(x)/(x^2))

{ "query": { "display": "$$\\frac{d}{dx}\\left(\\frac{\\sin\\left(x\\right)}{x^{2}}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dx}(\\frac{\\sin(x)}{x^{2}})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{x\\cos(x)-2\\sin(x)}{x^{3}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{\\sin\\left(x\\right)}{x^{2}}\\right)=\\frac{x\\cos\\left(x\\right)-2\\sin\\left(x\\right)}{x^{3}}$$", "input": "\\frac{d}{dx}\\left(\\frac{\\sin\\left(x\\right)}{x^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{d}{dx}\\left(\\sin\\left(x\\right)\\right)x^{2}-\\frac{d}{dx}\\left(x^{2}\\right)\\sin\\left(x\\right)}{\\left(x^{2}\\right)^{2}}" }, { "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": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "step", "result": "=\\frac{\\cos\\left(x\\right)x^{2}-2x\\sin\\left(x\\right)}{\\left(x^{2}\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{\\cos\\left(x\\right)x^{2}-2x\\sin\\left(x\\right)}{\\left(x^{2}\\right)^{2}}:{\\quad}\\frac{x\\cos\\left(x\\right)-2\\sin\\left(x\\right)}{x^{3}}$$", "input": "\\frac{\\cos\\left(x\\right)x^{2}-2x\\sin\\left(x\\right)}{\\left(x^{2}\\right)^{2}}", "result": "=\\frac{x\\cos\\left(x\\right)-2\\sin\\left(x\\right)}{x^{3}}", "steps": [ { "type": "interim", "title": "Factor $$\\cos\\left(x\\right)x^{2}-2x\\sin\\left(x\\right):{\\quad}x\\left(\\cos\\left(x\\right)x-2\\sin\\left(x\\right)\\right)$$", "input": "\\cos\\left(x\\right)x^{2}-2x\\sin\\left(x\\right)", "result": "=\\frac{x\\left(\\cos\\left(x\\right)x-2\\sin\\left(x\\right)\\right)}{\\left(x^{2}\\right)^{2}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=\\cos\\left(x\\right)xx-2x\\sin\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(\\cos\\left(x\\right)x-2\\sin\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\left(x^{2}\\right)^{2}:{\\quad}x^{4}$$", "input": "\\left(x^{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=x^{2\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jd2F9SNQ4l8Y5LjObCinw96GQqufR6tr2vPxOUv7H+9UXCH/RczT7MepT8JUGxNwZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz13rvDT7XPh5g1GffZc9C9/" } }, { "type": "step", "result": "=\\frac{x\\left(x\\cos\\left(x\\right)-2\\sin\\left(x\\right)\\right)}{x^{4}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{x\\cos\\left(x\\right)-2\\sin\\left(x\\right)}{x^{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Kqg/3+OyCpVPWw23luNHAjqloeWD2YI9549IjW/CMTLMmEFCCmQYAn+oFU3A3jfWLTrWWMFI8l4Q07DZ5+hJa0hqcfXty6JYd1pWIgQ3ZjjPLCW2ot4zlPfuMPpfddOGStKi4uvMsu7q3/6Vm+prlaN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwgrpVKVzXbsSNWy/mdIs9StinUW1qF9yFHykOlkM4yL3sJwR9sfdOtlIQSl2sFn8JJJLd1ohke2Wgml78++2zI0g==" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Quotient%20Rule", "practiceTopic": "Quotient Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{x\\cos(x)-2\\sin(x)}{x^{3}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }