derivative of xsqrt(1-x^2)

{ "query": { "display": "derivative of $$x\\sqrt{1-x^{2}}$$", "symbolab_question": "PRE_CALC#derivative x\\sqrt{1-x^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{-2x^{2}+1}{\\sqrt{1-x^{2}}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x\\sqrt{1-x^{2}}\\right)=\\frac{-2x^{2}+1}{\\sqrt{1-x^{2}}}$$", "input": "\\frac{d}{dx}\\left(x\\sqrt{1-x^{2}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=x,\\:g=\\sqrt{1-x^{2}}$$" ], "result": "=\\frac{dx}{dx}\\sqrt{1-x^{2}}+\\frac{d}{dx}\\left(\\sqrt{1-x^{2}}\\right)x", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{1-x^{2}}\\right)=-\\frac{x}{\\sqrt{1-x^{2}}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{1-x^{2}}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{1-x^{2}}}\\frac{d}{dx}\\left(1-x^{2}\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{1-x^{2}}\\right)", "result": "=\\frac{1}{2\\sqrt{1-x^{2}}}\\frac{d}{dx}\\left(1-x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=1-x^{2}$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(1-x^{2}\\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(1-x^{2}\\right)" }, { "type": "step", "primary": "Substitute back $$u=1-x^{2}$$", "result": "=\\frac{1}{2\\sqrt{1-x^{2}}}\\frac{d}{dx}\\left(1-x^{2}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgpPRx27ydfRxQ1Rvr5YtoAt9MMddrkFuNTLaPWMeYe3Z3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJXv++bSprT8DRLjDQza+XRVMvzzY1urVHAnTU92p3j+eqKwxODUNSw+sWQrjsE7uIFD/G+EpegrDq/zOyI9v/msRSpN33oxZMojoqvYhvSJAFuBNke0eZANmQMdPqVsU1Pq9PUUBMmxzxm22kIDr81B" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1-x^{2}\\right)=-2x$$", "input": "\\frac{d}{dx}\\left(1-x^{2}\\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(x^{2}\\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": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "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": "step", "result": "=0-2x" }, { "type": "step", "primary": "Simplify", "result": "=-2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{1-x^{2}}}\\left(-2x\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\sqrt{1-x^{2}}}\\left(-2x\\right):{\\quad}-\\frac{x}{\\sqrt{1-x^{2}}}$$", "input": "\\frac{1}{2\\sqrt{1-x^{2}}}\\left(-2x\\right)", "result": "=-\\frac{x}{\\sqrt{1-x^{2}}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2\\sqrt{1-x^{2}}}\\cdot\\:2x" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:2x}{2\\sqrt{1-x^{2}}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1\\cdot\\:x}{\\sqrt{1-x^{2}}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=-\\frac{x}{\\sqrt{-x^{2}+1}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13CwN3yDZpLrNdMYTkn2X9llaO6Dd3zpuSnKXNDXc0BYg4ebTKSUwQXHFFpEsrzHtUqUlp37JZxv9+CIGyLDYI+uDb2Sko4DkWORCZSWiJYDdkgQUxJPyUNnGfVirkcwpVO0BvnUFx4T5rmmmJUxss5NmDbMfNMThqBPZI/ssF14xUM8nwVBDyTe+B0GQFX12R+CS3daIZHtloJpe/PvtsyNI=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=1\\cdot\\:\\sqrt{1-x^{2}}+\\left(-\\frac{x}{\\sqrt{1-x^{2}}}\\right)x" }, { "type": "interim", "title": "Simplify $$1\\cdot\\:\\sqrt{1-x^{2}}+\\left(-\\frac{x}{\\sqrt{1-x^{2}}}\\right)x:{\\quad}\\frac{-2x^{2}+1}{\\sqrt{1-x^{2}}}$$", "input": "1\\cdot\\:\\sqrt{1-x^{2}}+\\left(-\\frac{x}{\\sqrt{1-x^{2}}}\\right)x", "result": "=\\frac{-2x^{2}+1}{\\sqrt{1-x^{2}}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=1\\cdot\\:\\sqrt{1-x^{2}}-\\frac{x}{\\sqrt{1-x^{2}}}x" }, { "type": "interim", "title": "$$1\\cdot\\:\\sqrt{1-x^{2}}=\\sqrt{1-x^{2}}$$", "input": "1\\cdot\\:\\sqrt{1-x^{2}}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sqrt{1-x^{2}}=\\sqrt{1-x^{2}}$$", "result": "=\\sqrt{1-x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UHFhuJ6yJNDOcGH7Fz25/6EeWqY4q/jIfEpqa0E5pW8DnzlbPZjyKgy1eUCFsLd5AsREamzBQPbK6HP8iuvD/KBtZyu+LaYvasa3tFjSwOE5S/rzXhIImTcNKflhJ16K0PXMJ/OaGc2pM8KTXwNIpQOFB0ieXqT2RpDCMcnacHQ=" } }, { "type": "interim", "title": "$$\\frac{x}{\\sqrt{1-x^{2}}}x=\\frac{x^{2}}{\\sqrt{1-x^{2}}}$$", "input": "\\frac{x}{\\sqrt{1-x^{2}}}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{xx}{\\sqrt{1-x^{2}}}" }, { "type": "interim", "title": "$$xx=x^{2}$$", "input": "xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74tglIeOAQqOR+OcPiyzQUczBWJotReR4P4m6RE6FZ2M7Aq6fHyeqJtW5OKbXVcT+IBF/biSmVq3Z2pV/8nBrAiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\frac{x^{2}}{\\sqrt{1-x^{2}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73Y7dsHR5Q/6nneLHJZRjGZ1i6bY7agL6zpEwNU8NfeDehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpmgkRqapkdrg4yiK5jAzXV1TENVM5lrkpLfucRPZru95mJW+vqX35fV9zI/bFAf9gIMTgbeeC0juQZMjBQOTgKLnHaVVCnUhcvup9B6HTXulgxOBt54LSO5BkyMFA5OAohZFQ8OchBdgxMOJrEMyk3s=" } }, { "type": "step", "result": "=\\sqrt{-x^{2}+1}-\\frac{x^{2}}{\\sqrt{-x^{2}+1}}" }, { "type": "step", "primary": "Convert element to fraction: $$\\sqrt{-x^{2}+1}=\\frac{\\sqrt{1-x^{2}}\\sqrt{1-x^{2}}}{\\sqrt{1-x^{2}}}$$", "result": "=-\\frac{x^{2}}{\\sqrt{1-x^{2}}}+\\frac{\\sqrt{1-x^{2}}\\sqrt{1-x^{2}}}{\\sqrt{1-x^{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{-x^{2}+\\sqrt{1-x^{2}}\\sqrt{1-x^{2}}}{\\sqrt{1-x^{2}}}" }, { "type": "interim", "title": "$$-x^{2}+\\sqrt{1-x^{2}}\\sqrt{1-x^{2}}=-2x^{2}+1$$", "input": "-x^{2}+\\sqrt{1-x^{2}}\\sqrt{1-x^{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{-x^{2}+1}\\sqrt{-x^{2}+1}=1-x^{2}$$" ], "result": "=-x^{2}+1-x^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Group like terms", "result": "=-x^{2}-x^{2}+1" }, { "type": "step", "primary": "Add similar elements: $$-x^{2}-x^{2}=-2x^{2}$$", "result": "=-2x^{2}+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jaxItqg/p85u4UD5o36ZlDenkeIeFESZctVD9yn7ngUb0QnaQXeEGehdJB+LYyEKCUCWbkwGOY7PqKo3U/JLJU07tC8Iy6HWHWTHo+CX7Hd8PILidLzxItlzEwp/VwP7tEiFW/r4jBylSnXHTO1/n8xTDkNqVjvz+cJmSUgwOoJBO1qW4wAPS8lxWIxIrigf" } }, { "type": "step", "result": "=\\frac{-2x^{2}+1}{\\sqrt{1-x^{2}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UHFhuJ6yJNDOcGH7Fz25/9ocVKOo8BB9GWXG6pqUdEqhs88jIycP+vJTm3llWr9A3da3b9zhsFMhk04lIzK4SXWD310L1+P2yDQQfMEhENFy81jD1KXXftuJY668njgtXHd474KY4N/MzsQUwOSVs+mIuOEdZZHc3tYsORarla9N5Aod6Hr1Lp2e/29KhSgUMEYLYt7kNh4Tw+x5qGopqfkY/WeMcHCW+o8If1dYfVqZsIuW0S7O0CHYANzKCPj91uo0mAhd9hm5EAEAzci8CLCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{-2x^{2}+1}{\\sqrt{1-x^{2}}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }