tangent of f(x)=1+ln(2x-1),\at x=1

{ "query": { "display": "tangent of $$f\\left(x\\right)=1+\\ln\\left(2x-1\\right),\\:\\at\\:x=1$$", "symbolab_question": "PRE_CALC#tangent f(x)=1+\\ln(2x-1),\\at x=1" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=2x-1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$f\\left(x\\right)=1+\\ln\\left(2x-1\\right)$$, at $$x=1:{\\quad}y=2x-1$$", "steps": [ { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(1,\\:1\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=1$$ into the equation $$f\\left(x\\right)=1+\\ln\\left(2x-1\\right)$$", "result": "f\\left(x\\right)=1+\\ln\\left(2\\cdot\\:1-1\\right)" }, { "type": "step", "primary": "Solve $$f\\left(x\\right)$$", "result": "f\\left(x\\right)=1" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$f\\left(x\\right)=1+\\ln\\left(2x-1\\right):{\\quad}\\frac{df\\left(x\\right)}{dx}=\\frac{2}{2x-1}$$", "input": "f\\left(x\\right)=1+\\ln\\left(2x-1\\right)", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$1+\\ln\\left(2x-1\\right)$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1+\\ln\\left(2x-1\\right)\\right)=\\frac{2}{2x-1}$$", "input": "\\frac{d}{dx}\\left(1+\\ln\\left(2x-1\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(1\\right)+\\frac{d}{dx}\\left(\\ln\\left(2x-1\\right)\\right)" }, { "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": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(2x-1\\right)\\right)=\\frac{2}{2x-1}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(2x-1\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2x-1}\\frac{d}{dx}\\left(2x-1\\right)$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(2x-1\\right)\\right)", "result": "=\\frac{1}{2x-1}\\frac{d}{dx}\\left(2x-1\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\ln\\left(u\\right),\\:\\:u=2x-1$$" ], "result": "=\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)\\frac{d}{dx}\\left(2x-1\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "input": "\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\ln\\left(u\\right)\\right)=\\frac{1}{u}$$", "result": "=\\frac{1}{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoqTCAmruKWcJsn66ZPDMT8cjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJq1PlbV5jLoKv9solFCc4blTW26qciuyUBGXQExCUedYd9mDo5FIvzrirtH7/W8pPUxk6YPA4jUd3Af4X0JJJ64=" } }, { "type": "step", "result": "=\\frac{1}{u}\\frac{d}{dx}\\left(2x-1\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x-1$$", "result": "=\\frac{1}{2x-1}\\frac{d}{dx}\\left(2x-1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjnW7cIPTknEQYBZ7ACWHnenttLfJY1eUK2Tk/4tHHiGurf72DA1fe1V1/79l0JjTJ1qmUmvlTu9Qd9Ei/4xn2twKjWO5SgJIqSpOp6JH5TXi9B9Su0yA/uoIhMYxVhrLCD11ILGJzXhggYTdnOU6b3WwPs1+Gw97t4MeuaNjSYTyjkVra0ajChSguMMf9fGqJUWlZ+bnpTem3dqAxhtNKM=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x-1\\right)=2$$", "input": "\\frac{d}{dx}\\left(2x-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(2x\\right)-\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw" } }, { "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": "=2-0" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2x-1}\\cdot\\:2" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2x-1}\\cdot\\:2:{\\quad}\\frac{2}{2x-1}$$", "input": "\\frac{1}{2x-1}\\cdot\\:2", "result": "=\\frac{2}{2x-1}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2x-1}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{2x-1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zTipe5NbYKWAN6+o+mf0TlXNWjXupFvjSy6Ui+S9W/kDnzlbPZjyKgy1eUCFsLd58xdtHZETggOffN1gbvOJ3WpPAJuNDKxWhHgKp3IU49GLGmNnLPWGf9PH3lpmjoJIWNsIC6hqeHCOj/1syn/vaeMNRtjIYm8lJ5fg9qNi5Sm/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=0+\\frac{2}{2x-1}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{2}{2x-1}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "\\frac{2}{2x-1}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS/Bc+UX+Om9RVveY+7e1/i5v8hjLbwvQtsKecUeJaHbeeKao7hW0voz3VcTgg+4km8zxxZKC51bsyhIHK30BCRG8tfRDsP3XL/ZRKgnlm2jjvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=2$$", "steps": [ { "type": "step", "primary": "Plug $$x=1$$ into the equation $$\\frac{2}{2x-1}$$", "result": "\\frac{2}{2\\cdot\\:1-1}" }, { "type": "interim", "title": "Simplify $$\\frac{2}{2\\cdot\\:1-1}:{\\quad}2$$", "input": "\\frac{2}{2\\cdot\\:1-1}", "result": "=2", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:1-1=1$$", "input": "2\\cdot\\:1-1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2-1" }, { "type": "step", "primary": "Subtract the numbers: $$2-1=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vJrxxPiSfL6+QIKCHbQn7d6GQqufR6tr2vPxOUv7H++P6Ubiv/bIrpol3G9QIK7h1SvY/eJGzvEmlW7hoPETDgROD+cqilt9fV3eVDyIDO0=" } }, { "type": "step", "result": "=\\frac{2}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jBZ1sa1jpVBU6rOuKqjnxlzyDhSnCMkZoLfrAty4UjR1g99dC9fj9sg0EHzBIRDRawWtIL+Wh7IhnhzFlElxPx429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJC4dk4xm3x5a0qC4PQ0tC9/r8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "result": "m=2" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$2$$ and passing through $$\\left(1,\\:1\\right):{\\quad}y=2x-1$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$2$$ and passing through $$\\left(1,\\:1\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=-1$$", "steps": [ { "type": "step", "primary": "Plug the slope $$2$$ into $$y=mx+b$$", "result": "y=2x+b" }, { "type": "step", "primary": "Plug in $$\\left(1,\\:1\\right)$$: $$\\quad\\:x=1,\\:y=1$$", "result": "1=2\\cdot\\:1+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$1=2\\cdot\\:1+b{\\quad:\\quad}b=-1$$", "input": "1=2\\cdot\\:1+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "2\\cdot\\:1+b=1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "2+b=1" }, { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "2+b=1", "result": "b=-1", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "2+b-2=1-2" }, { "type": "step", "primary": "Simplify", "result": "b=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=-1" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=2$$ and $$\\mathbf{b}=-1$$", "result": "y=2x-1" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=2x-1" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent f(x)=1+\\ln(2x-1),\\at x=1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }