normal of y= x/(x^2+1),\at x=-1

{ "query": { "display": "normal of $$y=\\frac{x}{x^{2}+1},\\:\\at\\:x=-1$$", "symbolab_question": "PRE_CALC#normal y=\\frac{x}{x^{2}+1},\\at x=-1" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Normal", "default": "x=-1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Normal line to $$y=\\frac{x}{x^{2}+1}$$, at $$x=-1:{\\quad}x=-1$$", "steps": [ { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(-1,\\:-\\frac{1}{2}\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=-1$$ into the equation $$y=\\frac{x}{x^{2}+1}$$", "result": "y=\\frac{-1}{\\left(-1\\right)^{2}+1}" }, { "type": "step", "primary": "Solve $$y$$", "result": "y=-\\frac{1}{2}" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$y=\\frac{x}{x^{2}+1}:{\\quad}\\frac{dy}{dx}=\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}$$", "input": "y=\\frac{x}{x^{2}+1}", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$\\frac{x}{x^{2}+1}$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{x}{x^{2}+1}\\right)=\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}$$", "input": "\\frac{d}{dx}\\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{dx}{dx}\\left(x^{2}+1\\right)-\\frac{d}{dx}\\left(x^{2}+1\\right)x}{\\left(x^{2}+1\\right)^{2}}" }, { "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": "$$\\frac{d}{dx}\\left(x^{2}+1\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(1\\right)" }, { "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": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\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+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "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", "interimType": "Derivatives" } }, { "type": "step", "result": "\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS2K+9mtmbSX+FdH0UbSASlRnKZhtGjBjBkP49kzv4yukXqsK+e1TiGj9RsLkU0NRYBssc7cdQyGeGPUKHi411ktdqNt+i2lCjhXdyI1f4oXLJrcePRQ/fYmViFAHQHwUFQ==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=0$$", "steps": [ { "type": "step", "primary": "Plug $$x=-1$$ into the equation $$\\frac{-x^{2}+1}{\\left(x^{2}+1\\right)^{2}}$$", "result": "\\frac{-\\left(-1\\right)^{2}+1}{\\left(\\left(-1\\right)^{2}+1\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{-\\left(-1\\right)^{2}+1}{\\left(\\left(-1\\right)^{2}+1\\right)^{2}}:{\\quad}0$$", "input": "\\frac{-\\left(-1\\right)^{2}+1}{\\left(\\left(-1\\right)^{2}+1\\right)^{2}}", "result": "=0", "steps": [ { "type": "interim", "title": "$$-\\left(-1\\right)^{2}+1=0$$", "input": "-\\left(-1\\right)^{2}+1", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "step", "result": "=-1+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+1=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X3Z0gSeOhd8EP4gOUHj8LN6GQqufR6tr2vPxOUv7H+9m4kp0IWw8nLX7mAEwRz12c9aAAKDohWHy8Ai+pheNS5BNqZJGS9+iP9MAxG0kaIA=" } }, { "type": "step", "result": "=\\frac{0}{\\left(\\left(-1\\right)^{2}+1\\right)^{2}}" }, { "type": "interim", "title": "$$\\left(\\left(-1\\right)^{2}+1\\right)^{2}=2^{2}$$", "input": "\\left(\\left(-1\\right)^{2}+1\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "step", "result": "=\\left(1+1\\right)^{2}" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s708QrD2x57reM7VPLw8k34VnyYRz18HvB+rp63mPitc+zsHBJV0oRhKqf7h8oBCgah0R4kHw1LArZyd/YFzVYTdFNGiYX6xZsyN5zHyMl1RrnmSJg7gfZC1mySRAhekY2" } }, { "type": "step", "result": "=\\frac{0}{2^{2}}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71hzAx3wk3v13ajUvtLnU1WmYaTmgDqciq0ZRL0em0iLU+25LRniRJ5kNid0AHeG8CUCWbkwGOY7PqKo3U/JLJSlz64FJ6ua0avQr8NXHg3KLGmNnLPWGf9PH3lpmjoJIFWPpL4Qe0FnKBJSzTcgMHVL3BYJhFR6StKUTLwjhrhiG17XnSEiwuN6ecXdPAiRz" } }, { "type": "step", "result": "m=0" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Compute the slope of the perpendicular line:$${\\quad}m_{p}=\\infty\\:$$", "steps": [ { "type": "step", "primary": "The perpendicular slope to $$0$$ is $$\\infty$$" }, { "type": "step", "result": "m_{p}=\\infty\\:" } ], "meta": { "interimType": "Line Equation Slope Perpendicular Infinity 0Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$\\infty\\:$$ and passing through $$\\left(-1,\\:-\\frac{1}{2}\\right):{\\quad}x=-1$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$\\infty\\:$$ and passing through $$\\left(-1,\\:-\\frac{1}{2}\\right)$$" }, { "type": "step", "primary": "Compute the $$y$$ intercept", "secondary": [ "The slope is $$\\infty$$, therefore the line equation is vertical.<br/>The line passes the point $$\\left(-1,\\:-\\frac{1}{2}\\right)$$, therefore the line intersects the $$x-$$axis for" ], "result": "x=-1" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "x=-1" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x}{x^{2}+1}", "displayFormula": "y=\\frac{x}{x^{2}+1}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=-1", "displayFormula": "x=-1", "attributes": { "color": "GRAY", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-1,-\\frac{1}{2})" ], 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