tangent of f(x)=5x-4x^2,(-1,-9)

{ "query": { "display": "tangent of $$f\\left(x\\right)=5x-4x^{2},\\:\\left(-1,\\:-9\\right)$$", "symbolab_question": "PRE_CALC#tangent f(x)=5x-4x^{2},(-1,-9)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=13x+4", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$f\\left(x\\right)=5x-4x^{2}$$, at $$\\left(-1,\\:-9\\right):{\\quad}y=13x+4$$", "steps": [ { "type": "interim", "title": "Find the slope of $$f\\left(x\\right)=5x-4x^{2}:{\\quad}\\frac{df\\left(x\\right)}{dx}=5-8x$$", "input": "f\\left(x\\right)=5x-4x^{2}", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$5x-4x^{2}$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5x-4x^{2}\\right)=5-8x$$", "input": "\\frac{d}{dx}\\left(5x-4x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(5x\\right)-\\frac{d}{dx}\\left(4x^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5x\\right)=5$$", "input": "\\frac{d}{dx}\\left(5x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=5\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=5\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=5", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmZ9X+BZ/JyYYY1hJF6v/r/ZGku9zFkxwe1dTH8vycb9dZwuGwx+eQcEBv+dY3CjTFNbbqpyK7JQEZdATEJR51hJPYdcyzxncaY/pkYJ4XZq" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x^{2}\\right)=8x$$", "input": "\\frac{d}{dx}\\left(4x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsGgzDnzN0IPmwhs7frKR8eTdaV09PMxEKZ9FieghTFwbuQyGyH7AL8DepgTwjAnNKN6Hv6MoTMtvtU0IQwXdn/V+9abCUkdC/28BK3pLujtM2vkiNOK25CPGkWDnxgF4w==" } }, { "type": "step", "result": "=5-8x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "5-8x" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWSx5TJmilpfrPdk2Aah885x1kS3dlcCKpQTQcheuut7MkpA7G2Ewdip1ZKoeRlZ0J78W+XjRMV37i6F1gjd/ZIwh+QEWY5Y9FbolzCUwY3XQZ" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=13$$", "steps": [ { "type": "step", "primary": "Plug $$x=-1$$ into the equation $$5-8x$$", "result": "5-8\\left(-1\\right)" }, { "type": "interim", "title": "Simplify $$5-8\\left(-1\\right):{\\quad}13$$", "input": "5-8\\left(-1\\right)", "result": "=13", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=5+8\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:1=8$$", "result": "=5+8" }, { "type": "step", "primary": "Add the numbers: $$5+8=13$$", "result": "=13" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rASzck+8CSzf+aMeAEAe9AOfOVs9mPIqDLV5QIWwt3l0yAk7piWbHtvP8qJMzefseqXxdc+rps1CUyb7fqI2GTVqmCwv3vkzbssHDdaNc8RhxN9zWbVekB8xk3JUOn1a" } }, { "type": "step", "result": "m=13" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$13$$ and passing through $$\\left(-1,\\:-9\\right):{\\quad}y=13x+4$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$13$$ and passing through $$\\left(-1,\\:-9\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=4$$", "steps": [ { "type": "step", "primary": "Plug the slope $$13$$ into $$y=mx+b$$", "result": "y=13x+b" }, { "type": "step", "primary": "Plug in $$\\left(-1,\\:-9\\right)$$: $$\\quad\\:x=-1,\\:y=-9$$", "result": "-9=13\\left(-1\\right)+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$-9=13\\left(-1\\right)+b{\\quad:\\quad}b=4$$", "input": "-9=13\\left(-1\\right)+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "13\\left(-1\\right)+b=-9" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "-13\\cdot\\:1+b=-9" }, { "type": "step", "primary": "Multiply the numbers: $$13\\cdot\\:1=13$$", "result": "-13+b=-9" }, { "type": "interim", "title": "Move $$13\\:$$to the right side", "input": "-13+b=-9", "result": "b=4", "steps": [ { "type": "step", "primary": "Add $$13$$ to both sides", "result": "-13+b+13=-9+13" }, { "type": "step", "primary": "Simplify", "result": "b=4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=4" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=13$$ and $$\\mathbf{b}=4$$", "result": "y=13x+4" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=13x+4" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent f(x)=5x-4x^{2},\\at (-1,-9)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }