tangent of f(x)=sin(2x)cos(x),\at x=pi

{ "query": { "display": "tangent of $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right),\\:\\at\\:x=π$$", "symbolab_question": "PRE_CALC#tangent f(x)=\\sin(2x)\\cos(x),\\at x=π" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=-2x+2π", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right)$$, at $$x=π:{\\quad}y=-2x+2π$$", "steps": [ { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(π,\\:0\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=π$$ into the equation $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right)$$", "result": "f\\left(x\\right)=\\sin\\left(2π\\right)\\cos\\left(π\\right)" }, { "type": "step", "primary": "Solve $$f\\left(x\\right)$$", "result": "f\\left(x\\right)=0" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right):{\\quad}\\frac{df\\left(x\\right)}{dx}=2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)$$", "input": "f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$\\sin\\left(2x\\right)\\cos\\left(x\\right)$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\cos\\left(x\\right)\\right)=2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\cos\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sin\\left(2x\\right),\\:g=\\cos\\left(x\\right)$$" ], "result": "=\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)\\cos\\left(x\\right)+\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)\\sin\\left(2x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)=\\cos\\left(2x\\right)\\cdot\\:2$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\cos\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)$$", "input": "\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)", "result": "=\\cos\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sin\\left(u\\right),\\:\\:u=2x$$" ], "result": "=\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)\\frac{d}{dx}\\left(2x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$", "result": "=\\cos\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgerJLn9ae0g0/tUjnRuL1v8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaugHqnJiEuQ8NpaCSOBx7rI4+YUX37Aa/AAEf1Hkty8FUcUM2sEdv7dIX0bKYOeE19OmgDIY2KBZfpU9cYqvCXz4=" } }, { "type": "step", "result": "=\\cos\\left(u\\right)\\frac{d}{dx}\\left(2x\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=\\cos\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYh6yUCRk/MnA93xfZq2OhuSQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEzREhQBxQGxynSTy6vjYFdXrqk0J72Q7Tc2GvO8AjiU6iGk8vIJisuT2N3pfkW1JpbGddanbms4cigCpjwsLZ/8wk+QLk42sL8qUHm7drm4ACS3daIZHtloJpe/PvtsyNI=" } }, { "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": "step", "result": "=\\cos\\left(2x\\right)\\cdot\\:2" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$", "input": "\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$", "result": "=-\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoTIPsH/5VFEfonU6bvi80j8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zA92cpyjnQxeYFWLLJRXAqymcxh5GfxfsNed5mphvPA8hRo8/W0Hq8WASQDG4Qt2SWMOnp3ra3sqxOu3sWEDlnw=" } }, { "type": "step", "result": "=\\cos\\left(2x\\right)\\cdot\\:2\\cos\\left(x\\right)+\\left(-\\sin\\left(x\\right)\\right)\\sin\\left(2x\\right)" }, { "type": "step", "primary": "Simplify", "result": "=2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS+weOnmaCIjfrej4kJXvYnJx0PMKptP0yG56gJLsJ8/pTeQKHeh69S6dnv9vSoUoFNhEAUjge2w+ZGLMk7E66nGBaeOua2bhsRNkXvK5/oyT5bjKQobNYFvQ3RZ+fWf9lw==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=-2$$", "steps": [ { "type": "step", "primary": "Plug $$x=π$$ into the equation $$2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)$$", "result": "2\\cos\\left(2π\\right)\\cos\\left(π\\right)-\\sin\\left(π\\right)\\sin\\left(2π\\right)" }, { "type": "interim", "title": "Simplify $$2\\cos\\left(2π\\right)\\cos\\left(π\\right)-\\sin\\left(π\\right)\\sin\\left(2π\\right):{\\quad}-2$$", "input": "2\\cos\\left(2π\\right)\\cos\\left(π\\right)-\\sin\\left(π\\right)\\sin\\left(2π\\right)", "result": "=-2", "steps": [ { "type": "interim", "title": "$$2\\cos\\left(2π\\right)\\cos\\left(π\\right)=-2$$", "input": "2\\cos\\left(2π\\right)\\cos\\left(π\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(2π\\right)=1$$", "input": "\\cos\\left(2π\\right)", "result": "=2\\cdot\\:1\\cdot\\:\\cos\\left(π\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(2π\\right)=\\cos\\left(0\\right)$$", "input": "\\cos\\left(2π\\right)", "result": "=\\cos\\left(0\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$2π$$ as $$2π+0$$", "result": "=\\cos\\left(2π+0\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\cos$$: $$\\cos\\left(x+2π\\right)=\\cos\\left(x\\right)$$", "secondary": [ "$$\\cos\\left(2π+0\\right)=\\cos\\left(0\\right)$$" ], "result": "=\\cos\\left(0\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(0\\right)=1$$", "input": "\\cos\\left(0\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=1" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=1" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "interim", "title": "Simplify $$\\cos\\left(π\\right):{\\quad}-1$$", "input": "\\cos\\left(π\\right)", "result": "=2\\cdot\\:1\\cdot\\:\\left(-1\\right)", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\cos\\left(π\\right)=\\left(-1\\right)$$", "secondary": [ "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" ], "result": "=-1" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-2\\cdot\\:1\\cdot\\:1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1\\cdot\\:1=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xufXhDkCP60yL8UVDWDDJQANE3P7ehounOdB47mIWgOrju+5Z51e/ZZSD3gRHwjBy++q5TspDrdoyXuh0DUdTuwHtPTZQgCTaLxxp2ffbNaH4O4IpfGeZyZOD8hwNTnX" } }, { "type": "step", "result": "=-2-\\sin\\left(π\\right)\\sin\\left(2π\\right)" }, { "type": "interim", "title": "$$\\sin\\left(π\\right)\\sin\\left(2π\\right)=0$$", "input": "\\sin\\left(π\\right)\\sin\\left(2π\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\sin\\left(π\\right):{\\quad}0$$", "input": "\\sin\\left(π\\right)", "result": "=0\\cdot\\:\\sin\\left(2π\\right)", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sin\\left(π\\right)=0$$", "secondary": [ "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" ], "result": "=0" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\sin\\left(2π\\right)=0$$", "input": "\\sin\\left(2π\\right)", "result": "=0\\cdot\\:0", "steps": [ { "type": "interim", "title": "$$\\sin\\left(2π\\right)=\\sin\\left(0\\right)$$", "input": "\\sin\\left(2π\\right)", "result": "=\\sin\\left(0\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$2π$$ as $$2π+0$$", "result": "=\\sin\\left(2π+0\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\sin$$: $$\\sin\\left(x+2π\\right)=\\sin\\left(x\\right)$$", "secondary": [ "$$\\sin\\left(2π+0\\right)=\\sin\\left(0\\right)$$" ], "result": "=\\sin\\left(0\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(0\\right)=0$$", "input": "\\sin\\left(0\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "primary": "Multiply the numbers: $$0\\cdot\\:0=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eGdu+WKTow/xGVQ4i5h4qXrVK9CGpzwCGNCiiTAhk65wkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX5S35qHLu5GNQPuB3apEJSVKJjcQy0Dt0Qol+oll+/fx" } }, { "type": "step", "result": "=-2-0" }, { "type": "step", "primary": "Simplify" }, { "type": "step", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq" } }, { "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 $$\\left(π,\\:0\\right):{\\quad}y=-2x+2π$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$-2$$ and passing through $$\\left(π,\\:0\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=2π$$", "steps": [ { "type": "step", "primary": "Plug the slope $$-2$$ into $$y=mx+b$$", "result": "y=\\left(-2\\right)x+b" }, { "type": "step", "primary": "Plug in $$\\left(π,\\:0\\right)$$: $$\\quad\\:x=π,\\:y=0$$", "result": "0=\\left(-2\\right)π+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$0=\\left(-2\\right)π+b{\\quad:\\quad}b=2π$$", "input": "0=\\left(-2\\right)π+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\left(-2\\right)π+b=0" }, { "type": "step", "result": "-2π+b=0" }, { "type": "interim", "title": "Move $$2π\\:$$to the right side", "input": "-2π+b=0", "result": "b=2π", "steps": [ { "type": "step", "primary": "Add $$2π$$ to both sides", "result": "-2π+b+2π=0+2π" }, { "type": "step", "primary": "Simplify", "result": "b=2π" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=2π" } ], "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 $$\\mathbf{b}=2π$$", "result": "y=-2x+2π" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=-2x+2π" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent f(x)=\\sin(2x)\\cos(x),\\at x=π" }, "showViewLarger": true } }, "meta": { "showVerify": true } }