tangent of x^2+y^2+2y=0,(0,-2)

{ "query": { "display": "tangent of $$x^{2}+y^{2}+2y=0,\\:\\left(0,\\:-2\\right)$$", "symbolab_question": "PRE_CALC#tangent x^{2}+y^{2}+2y=0,(0,-2)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=-2", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$x^{2}+y^{2}+2y=0$$, at $$\\left(0,\\:-2\\right):{\\quad}y=-2$$", "steps": [ { "type": "interim", "title": "Find the slope of $$x^{2}+y^{2}+2y=0:{\\quad}\\frac{dy}{dx}=-\\frac{x}{y+1}$$", "input": "x^{2}+y^{2}+2y=0", "steps": [ { "type": "step", "primary": "Use implicit differentiation to find the slope of $$x^{2}+y^{2}+2y=0$$" }, { "type": "interim", "title": "Implicit Derivative $$\\frac{dy}{dx}$$ of $$x^{2}+y^{2}+2y=0:{\\quad}-\\frac{x}{y+1}$$", "input": "x^{2}+y^{2}+2y=0", "steps": [ { "type": "step", "primary": "Treat $$y$$ as $$y\\left(x\\right)$$" }, { "type": "interim", "title": "Differentiate both sides:$${\\quad}2x+2y\\frac{dy}{dx}+2\\frac{dy}{dx}=0$$", "steps": [ { "type": "step", "primary": "Differentiate both sides of the equation with respect to $$x$$" }, { "type": "step", "result": "\\frac{d}{dx}\\left(x^{2}+y^{2}+2y\\right)=\\frac{d}{dx}\\left(0\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+y^{2}+2y\\right)=2x+2y\\frac{dy}{dx}+2\\frac{dy}{dx}$$", "input": "\\frac{d}{dx}\\left(x^{2}+y^{2}+2y\\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(y^{2}\\right)+\\frac{d}{dx}\\left(2y\\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": "$$\\frac{d}{dx}\\left(y^{2}\\right)=2y\\frac{dy}{dx}$$", "input": "\\frac{d}{dx}\\left(y^{2}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}2y\\frac{dy}{dx}$$", "input": "\\frac{d}{dx}\\left(y^{2}\\right)", "result": "=2y\\frac{dy}{dx}", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=u^{2},\\:\\:u=y$$" ], "result": "=\\frac{d}{du}\\left(u^{2}\\right)\\frac{dy}{dx}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(u^{2}\\right)=2u$$", "input": "\\frac{d}{du}\\left(u^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2u^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYr+VZUwBnLdzbS6DZQ+f4s+k3hxk9aCfAWodBRxXgUexMchyqTAJWrzJaDbnNcFsJUeCBKuYKgaNJ253gLI69U79qbCA2QqVmvm3jGRXZ2ppvbGT4j1utMEkCDH25m/vlQ==" } }, { "type": "step", "result": "=2u\\frac{dy}{dx}" }, { "type": "step", "primary": "Substitute back $$u=y$$", "result": "=2y\\frac{dy}{dx}" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjK+G35dCgQ5IQGiQCf8CJOk3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErdjG0obLzlrzhtVZTJegtKKXZnFgd1wgPOyrlPQ5EKbmFIG/qLOF4tVQpD96aiQBTKa2ghUiIjw31lfk8k2PrXIsIjaxJ4DvjTb2fbKjbvtlQ==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2y\\right)=2\\frac{dy}{dx}$$", "input": "\\frac{d}{dx}\\left(2y\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dy}{dx}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYk4ZzB5FT+NnlA3O4imeOUvZGku9zFkxwe1dTH8vycb96ewxPVuiEt8uM/ZWXzTRFfkeLJWDUJu4ZA3DtdyzfRCmcxh5GfxfsNed5mphvPA8a9wiFYgkMeoIFguqstGVGHULvWsVk7qE1ARqY3i+qJY=" } }, { "type": "step", "result": "=2x+2y\\frac{dy}{dx}+2\\frac{dy}{dx}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(0\\right)=0$$", "input": "\\frac{d}{dx}\\left(0\\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+idP5vLVrjUll6eMdYiebw9sKRrwhQNSsgmYcalhJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtp/saIEzR4+ujcRofteYd1" } }, { "type": "step", "result": "2x+2y\\frac{dy}{dx}+2\\frac{dy}{dx}=0" } ], "meta": { "interimType": "Differentiate Short 0Eq" } }, { "type": "interim", "title": "Isolate $$\\frac{dy}{dx}:{\\quad}\\frac{dy}{dx}=-\\frac{x}{y+1}$$", "input": "2x+2y\\frac{dy}{dx}+2\\frac{dy}{dx}=0", "steps": [ { "type": "step", "primary": "For convenience, write $$\\frac{dy}{dx}$$ as $$y^{^{\\prime}}$$", "result": "2x+2yy^{^{^{\\prime}}}+2y^{^{^{\\prime}}}=0" }, { "type": "interim", "title": "Isolate $$y^{^{\\prime}}:{\\quad}y^{^{\\prime}}=-\\frac{x}{y+1}$$", "input": "2x+2yy^{^{\\prime}}+2y^{^{\\prime}}=0", "result": "y^{^{\\prime}}=-\\frac{x}{y+1}", "steps": [ { "type": "interim", "title": "Move $$2x\\:$$to the right side", "input": "2x+2yy^{^{\\prime}}+2y^{^{\\prime}}=0", "result": "2yy^{^{\\prime}}+2y^{^{\\prime}}=-2x", "steps": [ { "type": "step", "primary": "Subtract $$2x$$ from both sides", "result": "2x+2yy^{^{^{\\prime}}}+2y^{^{^{\\prime}}}-2x=0-2x" }, { "type": "step", "primary": "Simplify", "result": "2yy^{^{^{\\prime}}}+2y^{^{^{\\prime}}}=-2x" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Factor $$2yy^{^{\\prime}}+2y^{^{\\prime}}:{\\quad}2y^{^{\\prime}}\\left(y+1\\right)$$", "input": "2yy^{^{\\prime}}+2y^{^{\\prime}}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2y^{^{^{\\prime}}}y+1\\cdot\\:2y^{^{^{\\prime}}}" }, { "type": "step", "primary": "Factor out common term $$2y^{^{\\prime}}$$", "result": "=2y^{^{^{\\prime}}}\\left(y+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "2y^{^{^{\\prime}}}\\left(y+1\\right)=-2x" }, { "type": "interim", "title": "Divide both sides by $$2\\left(y+1\\right)$$", "input": "2y^{^{\\prime}}\\left(y+1\\right)=-2x", "result": "y^{^{\\prime}}=-\\frac{x}{y+1}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2\\left(y+1\\right)$$", "result": "\\frac{2y^{^{^{\\prime}}}\\left(y+1\\right)}{2\\left(y+1\\right)}=\\frac{-2x}{2\\left(y+1\\right)}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2y^{^{\\prime}}\\left(y+1\\right)}{2\\left(y+1\\right)}=\\frac{-2x}{2\\left(y+1\\right)}", "result": "y^{^{\\prime}}=-\\frac{x}{y+1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2y^{^{\\prime}}\\left(y+1\\right)}{2\\left(y+1\\right)}:{\\quad}y^{^{\\prime}}$$", "input": "\\frac{2y^{^{\\prime}}\\left(y+1\\right)}{2\\left(y+1\\right)}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=\\frac{y^{^{^{\\prime}}}\\left(y+1\\right)}{y+1}" }, { "type": "step", "primary": "Cancel the common factor: $$y+1$$", "result": "=y^{^{^{\\prime}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75cNSpXMPZfVzY8+wmGDvGCEqNSKh6n+l16dlyx+8llen2yrlR6vo1NO4I/UtCsnazMFYmi1F5Hg/ibpEToVnY8QCbgB1aEXmSItV3PcK7PtFKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qiqpzT4W033W8eKO2tKasGyB6TiTPysh+DM0kzeO8xBcMAnOldiBW7bLXqfMPUcjyg==" } }, { "type": "interim", "title": "Simplify $$\\frac{-2x}{2\\left(y+1\\right)}:{\\quad}-\\frac{x}{y+1}$$", "input": "\\frac{-2x}{2\\left(y+1\\right)}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2x}{2\\left(y+1\\right)}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=-\\frac{x}{y+1}" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=-\\frac{x}{y+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/K7QcD7z0NH+A7I9pqFgeBA2/U4mT6bSfi/55NO4Pi9wkKGJWEPFPk38sdJMsyPI0ew9cn1w7OzvBprBa+nMnv8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUl+DxrDgqFoj/I0rlz5S49HT4oq3qaDZHmN8NbLr5NGCg=" } }, { "type": "step", "result": "y^{^{^{\\prime}}}=-\\frac{x}{y+1}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "interimType": "Generic Isolate 1Eq" } }, { "type": "step", "primary": "Write $$y^{^{\\prime}}\\:$$as $$\\frac{dy}{dx}$$", "result": "\\frac{dy}{dx}=-\\frac{x}{y+1}" } ], "meta": { "interimType": "Generic Isolate 1Eq" } }, { "type": "step", "result": "\\frac{dy}{dx}=-\\frac{x}{y+1}" } ], "meta": { "solvingClass": "ImplicitDiff", "interimType": "Implicit Differentiation Top 3Eq" } }, { "type": "step", "result": "-\\frac{x}{y+1}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS0lsioBmLxj4E6NURtUTPYnwPjJo8Fi+uo29zEltAtWE7WPeUTVcyBhR3SY906evgKD3kpjmeGFjKuDb6wrgPIn0RFP0wQUUKGTB6G8KtOnY" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=0$$", "steps": [ { "type": "step", "primary": "Plug $$x=0$$ into the equation $$-\\frac{x}{y+1}$$", "result": "-\\frac{0}{y+1}" }, { "type": "step", "primary": "Plug $$y=-2$$ into the equation $$-\\frac{x}{y+1}$$", "result": "-\\frac{0}{-2+1}" }, { "type": "interim", "title": "Simplify $$-\\frac{0}{-2+1}:{\\quad}0$$", "input": "-\\frac{0}{-2+1}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g3jS9/m7pGy12RZ0ADEhcY8XVXzSvmQsW7B12mCksg0E5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXSLY6UgkZLTmogyD31c+5GVQXGZzLhFYtl1LTneGXX4Fw==" } }, { "type": "step", "result": "m=0" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$0$$ and passing through $$\\left(0,\\:-2\\right):{\\quad}y=-2$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$0$$ and passing through $$\\left(0,\\:-2\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=-2$$", "steps": [ { "type": "step", "primary": "Plug the slope $$0$$ into $$y=mx+b$$", "result": "y=0\\cdot\\:x+b" }, { "type": "step", "primary": "Plug in $$\\left(0,\\:-2\\right)$$: $$\\quad\\:x=0,\\:y=-2$$", "result": "-2=0\\cdot\\:0+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$-2=0\\cdot\\:0+b{\\quad:\\quad}b=-2$$", "input": "-2=0\\cdot\\:0+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "0\\cdot\\:0+b=-2" }, { "type": "step", "primary": "Multiply the numbers: $$0\\cdot\\:0=0$$", "result": "0+b=-2" }, { "type": "step", "primary": "$$0+b=b$$", "result": "b=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=-2" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=0$$ and $$\\mathbf{b}=-2$$", "result": "y=-2" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=-2" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent x^{2}+y^{2}+2y=0,\\at (0,-2)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }