derivative of f(x)=(12)/x ,\at x=2

{ "query": { "display": "derivative of $$f\\left(x\\right)=\\frac{12}{x},\\:\\at\\:x=2$$", "symbolab_question": "PRE_CALC#derivative f(x)=\\frac{12}{x},\\at x=2" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivative at a Point", "default": "-3" }, "steps": { "type": "interim", "title": "$$-3$$", "steps": [ { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{12}{x}\\right)=-\\frac{12}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{12}{x}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=12\\frac{d}{dx}\\left(\\frac{1}{x}\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=12\\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": "=12\\left(-1\\cdot\\:x^{-1-1}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$12\\left(-1\\cdot\\:x^{-1-1}\\right):{\\quad}-\\frac{12}{x^{2}}$$", "input": "12\\left(-1\\cdot\\:x^{-1-1}\\right)", "result": "=-\\frac{12}{x^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-12\\cdot\\:1\\cdot\\:x^{-1-1}" }, { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-12\\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": "=12\\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\\:12}{x^{2}}" }, { "type": "step", "primary": "Refine", "result": "=-\\frac{12}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pPot7i8zfM8wV2UlzTVZAqljsoY5c8GRou6mxchSQfQJQJZuTAY5js+oqjdT8kslPzxtO6HllPKQGFEDY8ZUVG7mG3jsubzzTKmJlmQMe/QeNvb7k0sVmuwf19w9aD9N0AmtX5CjHB/D9t4r6g2D2TtXPAQwCS7UAHj6IgvYX5ckt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "Find the value of $$-\\frac{12}{x^{2}}$$ for $$x=2:{\\quad}-3$$", "steps": [ { "type": "step", "primary": "Plug $$x=2$$ into the equation $$-\\frac{12}{x^{2}}$$", "result": "-\\frac{12}{2^{2}}" }, { "type": "interim", "title": "Simplify $$-\\frac{12}{2^{2}}:{\\quad}-3$$", "input": "-\\frac{12}{2^{2}}", "result": "=-3", "steps": [ { "type": "interim", "title": "Cancel $$\\frac{12}{2^{2}}:{\\quad}3$$", "input": "\\frac{12}{2^{2}}", "result": "=-3", "steps": [ { "type": "interim", "title": "Factor $$12:{\\quad}2^{2}\\cdot\\:3$$", "steps": [ { "type": "step", "primary": "Factor $$12=2^{2}\\cdot\\:3$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}\\cdot\\:3}{2^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$2^{2}$$", "result": "=3" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtje/zIHwcpRshCql6jUKvrdd47a0hQ8flDbGsI5To1dNNwes1pEn8jD21yE6H99Wk3kCh3oevUunZ7/b0qFKBQj8UxsvvAFYSNtRTEGfD7fDq1hkJmPhVUWhTSHB7vsvg==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": 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