derivative of 5e^x+2/(\sqrt[3]{x})

{ "query": { "display": "$$\\frac{d}{dx}\\left(5e^{x}+\\frac{2}{\\sqrt[3]{x}}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dx}(5e^{x}+\\frac{2}{\\sqrt[3]{x}})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "5e^{x}-\\frac{2}{3x^{\\frac{4}{3}}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5e^{x}+\\frac{2}{\\sqrt[3]{x}}\\right)=5e^{x}-\\frac{2}{3x^{\\frac{4}{3}}}$$", "input": "\\frac{d}{dx}\\left(5e^{x}+\\frac{2}{\\sqrt[3]{x}}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(5e^{x}\\right)+\\frac{d}{dx}\\left(\\frac{2}{\\sqrt[3]{x}}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5e^{x}\\right)=5e^{x}$$", "input": "\\frac{d}{dx}\\left(5e^{x}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=5\\frac{d}{dx}\\left(e^{x}\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=5e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYpv4Yb3WedO0UZUAdwu+H0eTdaV09PMxEKZ9FieghTFw6k6wVv70LnWrAwpT+CN1n1roeUCC5gNxQc9h7CboxnNjDT5Dj/fM73/u0bafjbUvPc5dKL4ccGRij7LvJcm+uiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{2}{\\sqrt[3]{x}}\\right)=-\\frac{2}{3x^{\\frac{4}{3}}}$$", "input": "\\frac{d}{dx}\\left(\\frac{2}{\\sqrt[3]{x}}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{d}{dx}\\left(\\frac{1}{\\sqrt[3]{x}}\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=2\\frac{d}{dx}\\left(x^{-\\frac{1}{3}}\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2\\left(-\\frac{1}{3}x^{-\\frac{1}{3}-1}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$2\\left(-\\frac{1}{3}x^{-\\frac{1}{3}-1}\\right):{\\quad}-\\frac{2}{3x^{\\frac{4}{3}}}$$", "input": "2\\left(-\\frac{1}{3}x^{-\\frac{1}{3}-1}\\right)", "result": "=-\\frac{2}{3x^{\\frac{4}{3}}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-2\\cdot\\:\\frac{1}{3}x^{-\\frac{1}{3}-1}" }, { "type": "interim", "title": "$$x^{-\\frac{1}{3}-1}=x^{-\\frac{4}{3}}$$", "input": "x^{-\\frac{1}{3}-1}", "steps": [ { "type": "interim", "title": "Join $$-\\frac{1}{3}-1:{\\quad}-\\frac{4}{3}$$", "input": "-\\frac{1}{3}-1", "result": "=x^{-\\frac{4}{3}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=-\\frac{1\\cdot\\:3}{3}-\\frac{1}{3}" }, { "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\\:3-1}{3}" }, { "type": "interim", "title": "$$-1\\cdot\\:3-1=-4$$", "input": "-1\\cdot\\:3-1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=-3-1" }, { "type": "step", "primary": "Subtract the numbers: $$-3-1=-4$$", "result": "=-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yPXxmgPjHPGQ8EUShTFs81XTSum/z5kLpMzXS1UJIez3yWUnuS833V7NJZXZ/AglyCE30rzMlUAIVDyhseMBrjLBfl4ex0JeiEglU/97lVI=" } }, { "type": "step", "result": "=\\frac{-4}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72mCAahKKKXv7BelN+3mIkQwHhiDzHd1Twmbb3tvkM5dwkKGJWEPFPk38sdJMsyPIEY1esDjZr+Zpo8REGMlINf5qUhOazEJHGAynbCrZT0m7TU59Punj5mIVUCpScbF0NrrAAtPId0QdnEObUEydWBUWZkSQqDhmN0YCobaN0JM=" } }, { "type": "step", "result": "=-2\\cdot\\:\\frac{1}{3}x^{-\\frac{4}{3}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-\\frac{4}{3}}=\\frac{1}{x^{\\frac{4}{3}}}$$" ], "result": "=2\\cdot\\:\\frac{1}{3}\\cdot\\:\\frac{1}{x^{\\frac{4}{3}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=-\\frac{1\\cdot\\:1\\cdot\\:2}{3x^{\\frac{4}{3}}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1\\cdot\\:2=2$$", "result": "=-\\frac{2}{3x^{\\frac{4}{3}}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PZsRBLAZif21PgUSyOpVcgxD/G0+RTJBbzJqDAlWSdakRSJeTsSYpsg7oWOiXP/9cJChiVhDxT5N/LHSTLMjyE3kmX3OZvDjBsZqHgf6ozhj/PVBUlayNmSSIuqNWCz0ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0+T1TpsELCjy/spIVn62GMDEP8bT5FMkFvMmoMCVZJ1mHPRk6YdoqyFdts9sZpTbo=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=5e^{x}-\\frac{2}{3x^{\\frac{4}{3}}}" } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=5e^{x}-\\frac{2}{3x^{\\frac{4}{3}}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }