derivative of f(x)=-7/x ,\at x=-3

{ "query": { "display": "derivative of $$f\\left(x\\right)=-\\frac{7}{x},\\:\\at\\:x=-3$$", "symbolab_question": "PRE_CALC#derivative f(x)=-\\frac{7}{x},\\at x=-3" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivative at a Point", "default": "\\frac{7}{9}" }, "steps": { "type": "interim", "title": "$$\\frac{7}{9}$$", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-\\frac{7}{x}\\right)=\\frac{7}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(-\\frac{7}{x}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-7\\frac{d}{dx}\\left(\\frac{1}{x}\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=-7\\frac{d}{dx}\\left(x^{-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": "=-7\\left(-1\\cdot\\:x^{-1-1}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$-7\\left(-1\\cdot\\:x^{-1-1}\\right):{\\quad}\\frac{7}{x^{2}}$$", "input": "-7\\left(-1\\cdot\\:x^{-1-1}\\right)", "result": "=\\frac{7}{x^{2}}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=7\\cdot\\:1\\cdot\\:x^{-1-1}" }, { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=7\\cdot\\:1\\cdot\\:x^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-2}=\\frac{1}{x^{2}}$$" ], "result": "=7\\cdot\\:1\\cdot\\:\\frac{1}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{1\\cdot\\:7}{x^{2}}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{7}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EEeJa1Va9BfyLZJ8T3imlKljsoY5c8GRou6mxchSQfQJQJZuTAY5js+oqjdT8kslqDCV/0YwKT1O1GhRcGq4/Ok+KKUPvS9SYIBRdKMJESmY3ASC+aZqPN1DBWUUsybFNaHaxMupDf9l0/6OfNc+C4pfkDad9gN8tbyDX2I4UAk=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find the value of $$\\frac{7}{x^{2}}$$ for $$x=-3:{\\quad}\\frac{7}{9}$$", "steps": [ { "type": "step", "primary": "Plug $$x=-3$$ into the equation $$\\frac{7}{x^{2}}$$", "result": "\\frac{7}{\\left(-3\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{7}{\\left(-3\\right)^{2}}:{\\quad}\\frac{7}{9}$$", "input": "\\frac{7}{\\left(-3\\right)^{2}}", "result": "=\\frac{7}{9}", "steps": [ { "type": "interim", "title": "$$\\left(-3\\right)^{2}=3^{2}$$", "input": "\\left(-3\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-3\\right)^{2}=3^{2}$$" ], "result": "=3^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DiFM8Xt0cYQ3q7JWVRPZcM0ag8T1MwTer44+aCS/ZFA8nbDyNbrSnrFcItTWaPZwZtf025QjpwjfUn/kUwetFOiKbjfIuZ59Og03MnErgqY=" } }, { "type": "step", "result": "=\\frac{7}{3^{2}}" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\frac{7}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SkUxTeazMwJTJFIFhRx/j11iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPI58kzARhAv6QTxMGktxKXlKN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwghDsK7LsDvKPOefPyPwNZuFiVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "\\frac{7}{9}" } ], "meta": { "interimType": "Value At Point Top 2Eq" } }, { "type": "step", "result": "\\frac{7}{9}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{1}{x}", "displayFormula": "y=\\frac{1}{x}", "derivativeFormula": "-\\frac{1}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmax": 0 }, { "evalFormula": "y=\\frac{1}{x}", "displayFormula": "y=\\frac{1}{x}", "derivativeFormula": "-\\frac{1}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmin": 0 }, { "evalFormula": "x=0", "displayFormula": "y=\\frac{7}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "DASH", "labels": [ "\\mathrm{Vertical\\:Asymptote}" ], "isAsymptote": true } }, { "evalFormula": "y=0", "displayFormula": "y=\\frac{7}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "DASH", "labels": [ "\\mathrm{Horizontal\\:Asymptote}" ], "isAsymptote": true } }, { "evalFormula": "y=\\frac{7}{x^{2}}", "displayFormula": "y=\\frac{7}{x^{2}}", "derivativeFormula": "-\\frac{14}{x^{3}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmax": 0 }, { "evalFormula": "y=\\frac{7}{x^{2}}", "displayFormula": "y=\\frac{7}{x^{2}}", "derivativeFormula": "-\\frac{14}{x^{3}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmin": 0 }, { "evalFormula": "y=\\frac{1}{x}", "displayFormula": "y=\\frac{1}{x}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{7}{x^{2}}", "displayFormula": "y=\\frac{7}{x^{2}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-3,\\frac{7}{9})" ], "pointsDecimal": [ { "fst": -3, "snd": 0.7777777777777778 } ], "attributes": [ { "color": "BLACK", "labels": [ "\\mathrm{Derivative\\:value\\:at\\:point:}\\:\\frac{7}{9}" ], "labelTypes": [ "SOLUTION" ], "labelColors": [ "BLACK" ] } ] }, "functionChanges": [ { "origFormulaLatex": [ "x^{-1}" ], "finalFormulaLatex": [ "\\frac{1}{x}" ], "plotTitle": "x^{-1}, \\frac{7}{x^{2}}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -16.2, "xMax": 16.2, "yMin": -17.263555555555556, "yMax": 20.032444444444444 } } } } }