tangent of f(x)=sqrt(x^2+15),\at x=7

{ "query": { "display": "tangent of $$f\\left(x\\right)=\\sqrt{x^{2}+15},\\:\\at\\:x=7$$", "symbolab_question": "PRE_CALC#tangent f(x)=\\sqrt{x^{2}+15},\\at x=7" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=\\frac{7}{8}x+\\frac{15}{8}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$f\\left(x\\right)=\\sqrt{x^{2}+15}$$, at $$x=7:{\\quad}y=\\frac{7}{8}x+\\frac{15}{8}$$", "steps": [ { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(7,\\:8\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=7$$ into the equation $$f\\left(x\\right)=\\sqrt{x^{2}+15}$$", "result": "f\\left(x\\right)=\\sqrt{7^{2}+15}" }, { "type": "step", "primary": "Solve $$f\\left(x\\right)$$", "result": "f\\left(x\\right)=8" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$f\\left(x\\right)=\\sqrt{x^{2}+15}:{\\quad}\\frac{df\\left(x\\right)}{dx}=\\frac{x}{\\sqrt{x^{2}+15}}$$", "input": "f\\left(x\\right)=\\sqrt{x^{2}+15}", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$\\sqrt{x^{2}+15}$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{x^{2}+15}\\right)=\\frac{x}{\\sqrt{x^{2}+15}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x^{2}+15}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{x^{2}+15}}\\frac{d}{dx}\\left(x^{2}+15\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x^{2}+15}\\right)", "result": "=\\frac{1}{2\\sqrt{x^{2}+15}}\\frac{d}{dx}\\left(x^{2}+15\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=x^{2}+15$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(x^{2}+15\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\sqrt{u}\\right)=\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{d}{du}\\left(\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{du}\\left(u^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{2}u^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}u^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{1}{2}u^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{u}}", "steps": [ { "type": "interim", "title": "$$u^{\\frac{1}{2}-1}=u^{-\\frac{1}{2}}$$", "input": "u^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=u^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKe0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjMvOxDqXzE3/CFO0TFmffHAH2kDe5DGYTz3TrPquGdIhyukSOA/1RgMKO0TMhInPOMabdUggEogUL9RT7PNKh0VQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}u^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{u}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{u}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{u}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{u}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JOPQ2g2GS9EQptV8nckZSrH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJv2RkT96g5Q5jVbn5fyeQzwB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2rvsVWktCxa7XSYzIK90x3+aTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{u}}\\frac{d}{dx}\\left(x^{2}+15\\right)" }, { "type": "step", "primary": "Substitute back $$u=x^{2}+15$$", "result": "=\\frac{1}{2\\sqrt{x^{2}+15}}\\frac{d}{dx}\\left(x^{2}+15\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjasHTJ4Bo7yMP4qgzIlGETYln1bEVR48YFdDqNK7UBA1FQ1qG6fJuWh1wKHeWVTZR/eiQ2bs0wqnHz5TvUcL8GBIQdF1r4bkEczkVETb80bg2zHzTE4agT2SP7LBdeMVPtOgNkVfpovoD/ZPxyv+02hlmanBtqgRpCyiRNd9fLvIaTy8gmKy5PY3el+RbUmlsZ11qduazhyKAKmPCwtn/zCT5AuTjawvypQebt2ubgAJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+15\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+15\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(15\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(15\\right)=0$$", "input": "\\frac{d}{dx}\\left(15\\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+idP5vLVrjUll6eMdYiM9Mu4KSCCYpiNvsELyT1HZGku9zFkxwe1dTH8vycb9TbAOxT8wOTlsw5gGf+Hdr1NbbqpyK7JQEZdATEJR51hWrQ4Hd2sLdqfBisPzaXV7" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{x^{2}+15}}\\cdot\\:2x" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\sqrt{x^{2}+15}}\\cdot\\:2x:{\\quad}\\frac{x}{\\sqrt{x^{2}+15}}$$", "input": "\\frac{1}{2\\sqrt{x^{2}+15}}\\cdot\\:2x", "result": "=\\frac{x}{\\sqrt{x^{2}+15}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x}{2\\sqrt{x^{2}+15}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1\\cdot\\:x}{\\sqrt{x^{2}+15}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=\\frac{x}{\\sqrt{x^{2}+15}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13CxP9Vd2huwtk5hYr3vgWyz/WlHUBbWgzvAiuEyVhLd+SCUCWbkwGOY7PqKo3U/JLJXhIn+S6HGYn0Q5F2U1TesSYQuyUXlj7IrfPeOoILrZaZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz1zl0jd6iPDOR+4XyGvHO93E/1V3aG7C2TmFive+BbLP7XWB0XpeI+omvogvwTLVxI=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "\\frac{x}{\\sqrt{x^{2}+15}}" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWSzNmtswFK++8Juk+ifXemMYelp5mxjTAZ+6N39prfyW6qFKStRvo2EnGegTHxetYg84vE7cX7rnNzewb4iv/2F02HaGhMs78LtixV7CalQvFngy+M3CR2XaM9Z1iB3gEsA==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=\\frac{7}{8}$$", "steps": [ { "type": "step", "primary": "Plug $$x=7$$ into the equation $$\\frac{x}{\\sqrt{x^{2}+15}}$$", "result": "\\frac{7}{\\sqrt{7^{2}+15}}" }, { "type": "interim", "title": "Simplify $$\\frac{7}{\\sqrt{7^{2}+15}}:{\\quad}\\frac{7}{8}$$", "input": "\\frac{7}{\\sqrt{7^{2}+15}}", "result": "=\\frac{7}{8}", "steps": [ { "type": "interim", "title": "$$\\sqrt{7^{2}+15}=\\sqrt{64}$$", "input": "\\sqrt{7^{2}+15}", "steps": [ { "type": "step", "primary": "$$7^{2}=49$$", "result": "=\\sqrt{49+15}" }, { "type": "step", "primary": "Add the numbers: $$49+15=64$$", "result": "=\\sqrt{64}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s733WjE1Qc580XdXZpw0z5o1+rVXIYCl1jsDU19dobIgH9ovYKijQYhJDCbxu/nAOJyYnXZrWlI5xN2T+3PvKCr3Lmwp2GTieS/P9oLHcpsNQLF6AUdfeHvWRiCQSvFKRXelW0OPk1zi90GGxhyT9Wxg==" } }, { "type": "step", "result": "=\\frac{7}{\\sqrt{64}}" }, { "type": "interim", "title": "$$\\sqrt{64}=8$$", "input": "\\sqrt{64}", "steps": [ { "type": "step", "primary": "Factor the number: $$64=8^{2}$$", "result": "=\\sqrt{8^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{8^{2}}=8$$" ], "result": "=8", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kWvAGfhqCOMph0PmVu1IvCAn9lkDfZkicUGkO3EF+Io/bU11IvTSXwDE+bw+Zrq8zGQBGgzjhY9wy62Ypaxu7JoMc15KcAx+4DVyqRN7cNA=" } }, { "type": "step", "result": "=\\frac{7}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DSSvgEvn5DLnI32MZHMfGuNUyj+eqY9+te8AIKgPzBzehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpmG9S1RVEoEbqPiUnRql3o4eNvb7k0sVmuwf19w9aD9NwFkJl/hQGUOe3kPB8GwiQm72D1ki8XELza0wQzvmP4gYj+WkIn5QRVMZOFHRSL1G" } }, { "type": "step", "result": "m=\\frac{7}{8}" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$\\frac{7}{8}$$ and passing through $$\\left(7,\\:8\\right):{\\quad}y=\\frac{7}{8}x+\\frac{15}{8}$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$\\frac{7}{8}$$ and passing through $$\\left(7,\\:8\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=\\frac{15}{8}$$", "steps": [ { "type": "step", "primary": "Plug the slope $$\\frac{7}{8}$$ into $$y=mx+b$$", "result": "y=\\frac{7}{8}x+b" }, { "type": "step", "primary": "Plug in $$\\left(7,\\:8\\right)$$: $$\\quad\\:x=7,\\:y=8$$", "result": "8=\\frac{7}{8}\\cdot\\:7+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$8=\\frac{7}{8}\\cdot\\:7+b{\\quad:\\quad}b=\\frac{15}{8}$$", "input": "8=\\frac{7}{8}\\cdot\\:7+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\frac{7}{8}\\cdot\\:7+b=8" }, { "type": "interim", "title": "$$\\frac{7}{8}\\cdot\\:7=\\frac{49}{8}$$", "input": "\\frac{7}{8}\\cdot\\:7", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{7\\cdot\\:7}{8}" }, { "type": "step", "primary": "Multiply the numbers: $$7\\cdot\\:7=49$$", "result": "=\\frac{49}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WFDqwyIujo3AKt0Sxyd1AqUyhx2U/i+m6hs8FhqUqK+rju+5Z51e/ZZSD3gRHwjBMeCTlyHQDbsGiAsCczAWQT/L0MoYg+CUn6oyL3EO7YqZ1yxQg9bR7dNEGAE29vLgpinSf6Ksv/0SgnMx8E8oy8FGyAnbu3zZ2Xnj6PZ0lBY=" } }, { "type": "step", "result": "\\frac{49}{8}+b=8" }, { "type": "interim", "title": "Move $$\\frac{49}{8}\\:$$to the right side", "input": "\\frac{49}{8}+b=8", "result": "b=\\frac{15}{8}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{49}{8}$$ from both sides", "result": "\\frac{49}{8}+b-\\frac{49}{8}=8-\\frac{49}{8}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{49}{8}+b-\\frac{49}{8}=8-\\frac{49}{8}", "result": "b=\\frac{15}{8}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{49}{8}+b-\\frac{49}{8}:{\\quad}b$$", "input": "\\frac{49}{8}+b-\\frac{49}{8}", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\frac{49}{8}-\\frac{49}{8}=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$8-\\frac{49}{8}:{\\quad}\\frac{15}{8}$$", "input": "8-\\frac{49}{8}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$8=\\frac{8\\cdot\\:8}{8}$$", "result": "=\\frac{8\\cdot\\:8}{8}-\\frac{49}{8}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{8\\cdot\\:8-49}{8}" }, { "type": "interim", "title": "$$8\\cdot\\:8-49=15$$", "input": "8\\cdot\\:8-49", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:8=64$$", "result": "=64-49" }, { "type": "step", "primary": "Subtract the numbers: $$64-49=15$$", "result": "=15" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CtiZv+cLqca1q0VTuCGjqVXTSum/z5kLpMzXS1UJIeyHbzkedRNoxJTbODC7IXekVhjK2M4MjLj0NJtxn511x4zfBvttfooz49A+aJWvGWo=" } }, { "type": "step", "result": "=\\frac{15}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DSfc8u6USUlOydMbxepxe3yRHuGw7+tM5METTDj6vVFCyVMMrE5H/on5k4a9Rxq6Cd+3eUS+RnC+FVJ+td2GaIEFMST8lDZxn1Yq5HMKVTuU+GbS4DhG02Tt8DKQNLiGydpRdevcbLx7DIExCT8Q3w==" } }, { "type": "step", "result": "b=\\frac{15}{8}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=\\frac{15}{8}" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=\\frac{7}{8}$$ and $$\\mathbf{b}=\\frac{15}{8}$$", "result": "y=\\frac{7}{8}x+\\frac{15}{8}" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=\\frac{7}{8}x+\\frac{15}{8}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent f(x)=\\sqrt{x^{2}+15},\\at x=7" }, "showViewLarger": true } }, "meta": { "showVerify": true } }