derivative of y=arcsin(2x+1)

{ "query": { "display": "derivative of $$y=\\arcsin\\left(2x+1\\right)$$", "symbolab_question": "PRE_CALC#derivative y=\\arcsin(2x+1)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{1}{\\sqrt{-x(x+1)}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\arcsin\\left(2x+1\\right)\\right)=\\frac{1}{\\sqrt{-x\\left(x+1\\right)}}$$", "input": "\\frac{d}{dx}\\left(\\arcsin\\left(2x+1\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{\\sqrt{1-\\left(2x+1\\right)^{2}}}\\frac{d}{dx}\\left(2x+1\\right)$$", "input": "\\frac{d}{dx}\\left(\\arcsin\\left(2x+1\\right)\\right)", "result": "=\\frac{1}{\\sqrt{1-\\left(2x+1\\right)^{2}}}\\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": [ "$$f=\\arcsin\\left(u\\right),\\:\\:u=2x+1$$" ], "result": "=\\frac{d}{du}\\left(\\arcsin\\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": "$$\\frac{d}{du}\\left(\\arcsin\\left(u\\right)\\right)=\\frac{1}{\\sqrt{1-u^{2}}}$$", "input": "\\frac{d}{du}\\left(\\arcsin\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\arcsin\\left(u\\right)\\right)=\\frac{1}{\\sqrt{1-u^{2}}}$$", "result": "=\\frac{1}{\\sqrt{1-u^{2}}}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpRF3E3SKf3+BDVwL4fn4ZXyon5DWbx7ryBPXmhbAqLiqKwXnDyHJSOk7SW/uMHpmJCkbpQEWj02u4uNNwgketI1gv5YM8xCWZPrLqlF79LgDht/CJXqIwsnmwLWcvt05qnOC2uEox5qEsHQqHgPIyohmBKQUIpN5Cb2Y6XEDlLN3ooGiqzd+KtvCexJCsNgDbCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{1}{\\sqrt{1-u^{2}}}\\frac{d}{dx}\\left(2x+1\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x+1$$", "result": "=\\frac{1}{\\sqrt{1-\\left(2x+1\\right)^{2}}}\\frac{d}{dx}\\left(2x+1\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoNIWrQwDL8Ty+Lo4qSfhfIe90K/+aLLhaGQk94bxa/dqPySOWwYKYBMTeddEwOXMpq3Ed3KFeiHMOTCoq8eKidP0GDM0RHsMmIZkaA2q+6HPecntSMQYtNsD1uraIP3YLOEC3P0xhies8V/3p2YQO4a+EPD9Is1Ssmpf6WYy3HhRSpN33oxZMojoqvYhvSJAFuBNke0eZANmQMdPqVsU1Pq9PUUBMmxzxm22kIDr81B" } }, { "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}{\\sqrt{1-\\left(2x+1\\right)^{2}}}\\cdot\\:2" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sqrt{1-\\left(2x+1\\right)^{2}}}\\cdot\\:2:{\\quad}\\frac{1}{\\sqrt{-x\\left(x+1\\right)}}$$", "input": "\\frac{1}{\\sqrt{1-\\left(2x+1\\right)^{2}}}\\cdot\\:2", "result": "=\\frac{1}{\\sqrt{-x\\left(x+1\\right)}}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sqrt{1-\\left(2x+1\\right)^{2}}}=\\frac{1}{2\\sqrt{-x\\left(x+1\\right)}}$$", "input": "\\frac{1}{\\sqrt{1-\\left(2x+1\\right)^{2}}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{1-\\left(2x+1\\right)^{2}}=2\\sqrt{-x\\left(x+1\\right)}$$", "input": "\\sqrt{1-\\left(2x+1\\right)^{2}}", "steps": [ { "type": "interim", "title": "Expand $$1-\\left(2x+1\\right)^{2}:{\\quad}-4x^{2}-4x$$", "input": "1-\\left(2x+1\\right)^{2}", "result": "=\\sqrt{-4x^{2}-4x}", "steps": [ { "type": "interim", "title": "$$\\left(2x+1\\right)^{2}:{\\quad}4x^{2}+4x+1$$", "result": "=1-\\left(4x^{2}+4x+1\\right)", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$", "secondary": [ "$$a=2x,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=\\left(2x\\right)^{2}+2\\cdot\\:2x\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$\\left(2x\\right)^{2}+2\\cdot\\:2x\\cdot\\:1+1^{2}:{\\quad}4x^{2}+4x+1$$", "input": "\\left(2x\\right)^{2}+2\\cdot\\:2x\\cdot\\:1+1^{2}", "result": "=4x^{2}+4x+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\left(2x\\right)^{2}+2\\cdot\\:2\\cdot\\:1\\cdot\\:x+1" }, { "type": "interim", "title": "$$\\left(2x\\right)^{2}=4x^{2}$$", "input": "\\left(2x\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}x^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7A9Ss89QDrwVf91N0ewnmtM0ag8T1MwTer44+aCS/ZFDRZIoeY9W2mCwZ4DKHjHhjdCAW7st8OLhZc81DEkXmUHbEM+STIyl9P3F0FTKS2/k=" } }, { "type": "interim", "title": "$$2\\cdot\\:2x\\cdot\\:1=4x$$", "input": "2\\cdot\\:2x\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2\\cdot\\:1=4$$", "result": "=4x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CX40xexqpVGsxUm3TFBpSGBFH3ZqAOJQyqKKX506iE3MwViaLUXkeD+JukROhWdjhJ2chl2xWYW8EO6x3Dp1jHE95UF9K86ID4Pc5O/k1J3ztcMh3l1pOgbti4SxCC26" } }, { "type": "step", "result": "=4x^{2}+4x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$-\\left(4x^{2}+4x+1\\right):{\\quad}-4x^{2}-4x-1$$", "input": "-\\left(4x^{2}+4x+1\\right)", "result": "=1-4x^{2}-4x-1", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(4x^{2}\\right)-\\left(4x\\right)-\\left(1\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-4x^{2}-4x-1" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Simplify $$1-4x^{2}-4x-1:{\\quad}-4x^{2}-4x$$", "input": "1-4x^{2}-4x-1", "result": "=-4x^{2}-4x", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-4x^{2}-4x+1-1" }, { "type": "step", "primary": "$$1-1=0$$", "result": "=-4x^{2}-4x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77OJs7GggUlcwySL2Vt+371XTSum/z5kLpMzXS1UJIeykU1KMwaqxb2spciI7grUKvhMjRi/3rIor0bj5QFhnZxJyf8zawtgEaDEKWrMLEzbSPKXznDueg0O3Q6bsTIJFjwE87HTCWyAU3ypRroDMDQ==" } }, { "type": "interim", "title": "Factor $$-4x^{2}-4x:{\\quad}-4x\\left(x+1\\right)$$", "input": "-4x^{2}-4x", "result": "=\\sqrt{-4x\\left(x+1\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=-4xx-4x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Rewrite as", "result": "=-4xx-1\\cdot\\:4x" }, { "type": "step", "primary": "Factor out common term $$4x$$", "result": "=-4x\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\sqrt{4}\\sqrt{-x\\left(x+1\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{4}=2$$", "input": "\\sqrt{4}", "result": "=2\\sqrt{-x\\left(x+1\\right)}", "steps": [ { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vYTKhRE+LXcj9vBsHSzw2QbeC2QCdLxKEECRBZFZMUSrju+5Z51e/ZZSD3gRHwjBtlsC9OAlQvbN08y3yr7zVO4oEELu9g6W4CQkjIgRoSS04tHsGiX14BPznfJA3Rd3Hosf/qbmjHr2r4TJX39fRZsmdcrR2jNSuFpOKHXU+0o=" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{-x\\left(x+1\\right)}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGdKnLD7Y6tlGZm9T7wX6NaN8kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcauoNsx80xOGoE9kj+ywXXjFQi0TjiMzqYQ4S5bm386yTfU1tuqnIrslARl0BMQlHnWKFRe6XFTlnZqoH+DQfd7Gr4aYCYqijbLrXlH6Bsua/18U00AKCrXq5XsedL0BzvSCg9eelN04h59itH6Vvfb/A=" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{2\\sqrt{-x\\left(x+1\\right)}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2\\sqrt{-x\\left(x+1\\right)}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{\\sqrt{-x\\left(x+1\\right)}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGdKnLD7Y6tlGZm9T7wX6NaNVzVo17qRb40sulIvkvVv5A585Wz2Y8ioMtXlAhbC3ecVwzg7RQQgvFH/N+iHBPcd8x9vynyQ7JOGYnN81pxc4P8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXTdWZmqauxOWR0VplsqF0PtuwPiqydgIAabWE8heHnH1AoVhN9TPkl+OLhnsaXMlGE=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{1}{\\sqrt{-x(x+1)}}", "displayFormula": "y=\\frac{1}{\\sqrt{-x(x+1)}}", "derivativeFormula": "-\\frac{-2x-1}{2(-x(x+1))^{\\frac{3}{2}}}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false }, "xmin": -1, "xmax": 0, "calculatePoints": true }, { 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