d/(dy)(1/y)

{ "query": { "display": "$$\\frac{d}{dy}\\left(\\frac{1}{y}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dy}(\\frac{1}{y})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-\\frac{1}{y^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dy}\\left(\\frac{1}{y}\\right)=-\\frac{1}{y^{2}}$$", "input": "\\frac{d}{dy}\\left(\\frac{1}{y}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=\\frac{d}{dy}\\left(y^{-1}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=-1\\cdot\\:y^{-1-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$-1\\cdot\\:y^{-1-1}:{\\quad}-\\frac{1}{y^{2}}$$", "input": "-1\\cdot\\:y^{-1-1}", "result": "=-\\frac{1}{y^{2}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-1\\cdot\\:y^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$y^{-2}=\\frac{1}{y^{2}}$$" ], "result": "=-1\\cdot\\:\\frac{1}{y^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{y^{2}}=\\frac{1}{y^{2}}$$", "result": "=-\\frac{1}{y^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OgqB1bSeU1hnIoKXsDD6iKiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrTUiCiuhHNgfFa0MLwy3yR/8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUlVlzVbd8nUeV35Ms93fuQrBF1+E4wvPRIGnJs5KwUnrw=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=-\\frac{1}{x^{2}}", "displayFormula": "y=-\\frac{1}{x^{2}}", "derivativeFormula": "\\frac{2}{y^{3}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmax": 0, "calculatePoints": true }, { "evalFormula": "y=-\\frac{1}{x^{2}}", "displayFormula": "y=-\\frac{1}{x^{2}}", "derivativeFormula": "\\frac{2}{y^{3}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmin": 0, "calculatePoints": true }, { "evalFormula": "x=0", "displayFormula": "y=-\\frac{1}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "DASH", "labels": [ "\\mathrm{Vertical\\:Asymptote}" ], "isAsymptote": true } }, { "evalFormula": "y=0", "displayFormula": "y=-\\frac{1}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "DASH", "labels": [ "\\mathrm{Horizontal\\:Asymptote}" ], "isAsymptote": true } } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "y=-\\frac{1}{x^{2}}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -9.112499999999999, "xMax": 9.112499999999999, "yMin": -9.04463100137174, "yMax": 7.794480109739368 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }