derivative of csc(sqrt(x+5))

{ "query": { "display": "derivative of $$\\csc\\left(\\sqrt{x+5}\\right)$$", "symbolab_question": "PRE_CALC#derivative \\csc(\\sqrt{x+5})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-\\frac{\\cot(\\sqrt{x+5})\\csc(\\sqrt{x+5})}{2\\sqrt{x+5}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\csc\\left(\\sqrt{x+5}\\right)\\right)=-\\frac{\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)}{2\\sqrt{x+5}}$$", "input": "\\frac{d}{dx}\\left(\\csc\\left(\\sqrt{x+5}\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}-\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)$$", "input": "\\frac{d}{dx}\\left(\\csc\\left(\\sqrt{x+5}\\right)\\right)", "result": "=-\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\csc\\left(u\\right),\\:\\:u=\\sqrt{x+5}$$" ], "result": "=\\frac{d}{du}\\left(\\csc\\left(u\\right)\\right)\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\csc\\left(u\\right)\\right)=-\\cot\\left(u\\right)\\csc\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\csc\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\csc\\left(u\\right)\\right)=-\\cot\\left(u\\right)\\csc\\left(u\\right)$$", "result": "=-\\cot\\left(u\\right)\\csc\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsB0VBe8CtPscmwZuOSILXH8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zCEoZoBjo4XmXlwMYlZgwaNkS3dlcCKpQTQcheuut7Mkr7Lkk3W8UGM01WDfrx5SOuQZKSyCSXMun55LI4MdL+/JaUEWnAodameMJ/HHQDeX" } }, { "type": "step", "result": "=-\\cot\\left(u\\right)\\csc\\left(u\\right)\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\sqrt{x+5}$$", "result": "=-\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkK64/5BHjfS8ABwgxcMAhShIHnjQf85uv/pVuJHd2gILI71+ylVDvwHghUiHeEt94lsPYObl85kuSl+cyMzh2V4l9+XUP4dNZSL8WpmbbeTk3Am9bgdsurvCPhXwvyPUjZ3t4e0mUtO+THZyFy73boYyx3wUxkgWPZ7RfgIKpkaevBca1QUtXPodlxAiT57UKN6Hv6MoTMtvtU0IQwXdn8xqzBrmghqApAU5+ucqbn4Ztv8/FfdrC63krFHHJl/wA==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)=\\frac{1}{2\\sqrt{x+5}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{x+5}}\\frac{d}{dx}\\left(x+5\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x+5}\\right)", "result": "=\\frac{1}{2\\sqrt{x+5}}\\frac{d}{dx}\\left(x+5\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=x+5$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(x+5\\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+5\\right)" }, { "type": "step", "primary": "Substitute back $$u=x+5$$", "result": "=\\frac{1}{2\\sqrt{x+5}}\\frac{d}{dx}\\left(x+5\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjasHTJ4Bo7yMP4qgzIlGESSTsnUSJ5dVXipoSu3Ru0GdLl7DeVd7l7l/uUT/v1GhNJLZOXSTCXkMFKW90A2Pi/484O9NJ/PDg7/ATYMA16bkuys1a2jMgSUA48CoEjdh+/tLl11PD5ADADWLg4MvebLBSk1lkIRVqICk1aV2j0m3SmUvX4+ZxjvBdfdSGCsOwpcB1b+ayqTG1ksTw+VIa6/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x+5\\right)=1$$", "input": "\\frac{d}{dx}\\left(x+5\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}+\\frac{d}{dx}\\left(5\\right)" }, { "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": "$$\\frac{d}{dx}\\left(5\\right)=0$$", "input": "\\frac{d}{dx}\\left(5\\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+idP5vLVrjUll6eMdYmXEh6/dOKVl5+UiJ6t4qwxJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTvz/OzRy6l5fd6++0L3aMbw" } }, { "type": "step", "result": "=1+0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{x+5}}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2\\sqrt{x+5}}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=-\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)\\frac{1}{2\\sqrt{x+5}}" }, { "type": "interim", "title": "Simplify $$-\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)\\frac{1}{2\\sqrt{x+5}}:{\\quad}-\\frac{\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)}{2\\sqrt{x+5}}$$", "input": "-\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)\\frac{1}{2\\sqrt{x+5}}", "result": "=-\\frac{\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)}{2\\sqrt{x+5}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)}{2\\sqrt{x+5}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cot\\left(\\sqrt{x+5}\\right)=\\cot\\left(\\sqrt{x+5}\\right)$$", "result": "=-\\frac{\\cot\\left(\\sqrt{x+5}\\right)\\csc\\left(\\sqrt{x+5}\\right)}{2\\sqrt{x+5}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rC6vR+mKVm6SGuD9vQlsN2eVRAG9zmgMEX3E5FljVgFtWQN7gF9Lkc1CCtXSpzQhsIM3hsvXB4uOlb+gHeVLE1XTSum/z5kLpMzXS1UJIeyGAWQxLIaGRwm59Wqwx6r0cpWINwL67TnFeuON7bkeS2eVRAG9zmgMEX3E5FljVgFieYNK1XpaVngQmlYW0geitLeI4VTzQuqNBFAxjT1joh429vuTSxWa7B/X3D1oP00dw28oTTPhM/icPj7CUDbJ+JD/4FSCt8Rv+i9E1SiEjR2c15HYJv7RudcAjm38NamDbMfNMThqBPZI/ssF14xUlO4eJ27tyx4qScWTimE1BQ==" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{\\cot(\\sqrt{x+5})\\csc(\\sqrt{x+5})}{2\\sqrt{x+5}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }