(\partial)/(\partial x)(x/(x^2+1))

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x}{x^{2}+1}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial x}(\\frac{x}{x^{2}+1})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "\\frac{-x^{2}+1}{(x^{2}+1)^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x}{x^{2}+1}\\right)=\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(\\frac{x}{x^{2}+1}\\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{\\partial\\:}{\\partial\\:x}\\left(x\\right)\\left(x^{2}+1\\right)-\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{2}+1\\right)x}{\\left(x^{2}+1\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{\\partial\\:}{\\partial\\:x}\\left(x\\right)=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmuHQGTAre0/umYO3/E+LF4lyEB4JYjIUjkjbDZ4tfSJ+yeROYBotscHIZETI6FSe7NWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CdhSH/V18j9Kf/3yKXdVwr8kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{2}+1\\right)=2x$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{2}\\right)+\\frac{\\partial\\:}{\\partial\\:x}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(x^{2}\\right)=2x$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAlhtNDG2E/X19/N12BymK4vHI5S0StY1FdtOqqOPr0Te9RTuRyuNfmLd3VuZGWIfzJkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089utE82uh6KQgoV9se9/H1S1+unNN16jS33d19pgzn+cG" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:x}\\left(1\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:x}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAkJQloh76DUeoB3JfCRc1JYlyEB4JYjIUjkjbDZ4tfSJ3+y6gfQnMr2Alg7BrHl9PbNWyGcX6HZt1LGXH2QGa+Ln0ClXHqmT3uOusVLMnE4CTYJGKukFdo/SOyITpAF5ackt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1\\cdot\\:\\left(x^{2}+1\\right)-2xx}{\\left(x^{2}+1\\right)^{2}}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{2}+1\\right)-2xx=-x^{2}+1$$", "input": "1\\cdot\\:\\left(x^{2}+1\\right)-2xx", "result": "=\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$1\\cdot\\:\\left(x^{2}+1\\right)=x^{2}+1$$", "input": "1\\cdot\\:\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(x^{2}+1\\right)=\\left(x^{2}+1\\right)$$", "result": "=\\left(x^{2}+1\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=x^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HNREWIJAGttVJ8tLuFhXKz2gYtQEl4V79ys4pDeUXjWjkVi15I8rBefLi4Iyt2wrZfipvhu260EBw0vyu63vxk1KmoOKsbKJCeuDsUvATG/wKYd15lciiD0rY3SmLhdAsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$2xx=2x^{2}$$", "input": "2xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=2x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74UAakB14Lbm7SHYIpLTwTnCQoYlYQ8U+Tfyx0kyzI8iSVveXeWzQO/GlTVao5UKXszTt6qIJZczvODM49/dKgo8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "result": "=x^{2}+1-2x^{2}" }, { "type": "step", "primary": "Group like terms", "result": "=x^{2}-2x^{2}+1" }, { "type": "step", "primary": "Add similar elements: $$x^{2}-2x^{2}=-x^{2}$$", "result": "=-x^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HNREWIJAGttVJ8tLuFhXK54kF2IpnZAO71yKpjphLq8JQJZuTAY5js+oqjdT8ksl83iFzgC1jRav6XSHifJOjYpiDBqHg2RlromhyTcRK5hXdYO2Wwjmwk/VR33gpcrJoUMwwlVgJk5w51zpcjzcyA==" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }