derivative of f(x)=e^{1/x}

{ "query": { "display": "derivative of $$f\\left(x\\right)=e^{\\frac{1}{x}}$$", "symbolab_question": "PRE_CALC#derivative f(x)=e^{\\frac{1}{x}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-\\frac{e^{\\frac{1}{x}}}{x^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{\\frac{1}{x}}\\right)=-\\frac{e^{\\frac{1}{x}}}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(e^{\\frac{1}{x}}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{\\frac{1}{x}}\\frac{d}{dx}\\left(\\frac{1}{x}\\right)$$", "input": "\\frac{d}{dx}\\left(e^{\\frac{1}{x}}\\right)", "result": "=e^{\\frac{1}{x}}\\frac{d}{dx}\\left(\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=\\frac{1}{x}$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(\\frac{1}{x}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{d}{du}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqCr3EWRZw3L4+rHTTdVG0Ok3hxk9aCfAWodBRxXgUexwx+RE9MtjN5hKMwTI7fffj/L0MoYg+CUn6oyL3EO7YrHahlpzKGY893KZ4T4i4Tv3RCXWsqiNx7T9zOhL5sYfw==" } }, { "type": "step", "result": "=e^{u}\\frac{d}{dx}\\left(\\frac{1}{x}\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\frac{1}{x}$$", "result": "=e^{\\frac{1}{x}}\\frac{d}{dx}\\left(\\frac{1}{x}\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkHoFimzo2hAH54hEde9aCqpCAYm9WaodeZmcK0pGSvo1FQ1qG6fJuWh1wKHeWVTZR/eiQ2bs0wqnHz5TvUcL8E7MaqYzSQoXW0LdoOtO3Il3H93+PO0lEb6MrrW3QQq1QPvyOowd1nXwXCJho+3aCMfjoxxUZf9VHsm55vnQ+kzeqXxdc+rps1CUyb7fqI2Ga++0UXCFaDRjBhGCE8EEfmkQ/JnsM45+j77dquc6L38" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{1}{x}\\right)=-\\frac{1}{x^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{1}{x}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a}=a^{-1}$$", "result": "=\\frac{d}{dx}\\left(x^{-1}\\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": "=-1\\cdot\\:x^{-1-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$-1\\cdot\\:x^{-1-1}:{\\quad}-\\frac{1}{x^{2}}$$", "input": "-1\\cdot\\:x^{-1-1}", "result": "=-\\frac{1}{x^{2}}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=-1\\cdot\\:x^{-2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-2}=\\frac{1}{x^{2}}$$" ], "result": "=-1\\cdot\\:\\frac{1}{x^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{1}{x^{2}}=\\frac{1}{x^{2}}$$", "result": "=-\\frac{1}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+AgNZzeIicTfbr51/JGq1KiJTJQkRRngZl07rbqjeC6jkVi15I8rBefLi4Iyt2wrT1r2iaXj0z6hiHFOta70Gf8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUl/dfOQfCkILEECUyPIy9DrRF1+E4wvPRIGnJs5KwUnrw=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{\\frac{1}{x}}\\left(-\\frac{1}{x^{2}}\\right)" }, { "type": "interim", "title": "Simplify $$e^{\\frac{1}{x}}\\left(-\\frac{1}{x^{2}}\\right):{\\quad}-\\frac{e^{\\frac{1}{x}}}{x^{2}}$$", "input": "e^{\\frac{1}{x}}\\left(-\\frac{1}{x^{2}}\\right)", "result": "=-\\frac{e^{\\frac{1}{x}}}{x^{2}}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-e^{\\frac{1}{x}}\\frac{1}{x^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:e^{\\frac{1}{x}}}{x^{2}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{\\frac{1}{x}}=e^{\\frac{1}{x}}$$", "result": "=-\\frac{e^{\\frac{1}{x}}}{x^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GykIPI5T3FowFX0LyeV3R1SEDL5s0pN2jNrFECfEqArFdAes7PIXIofHuuahyaVXcJChiVhDxT5N/LHSTLMjyPPmmEPX/L8QYtGflEE2JbqkdIU6sukspiVugGNS0wcqAfaQN7kMZhPPdOs+q4Z0iB4pgUWEah0lniZLlD4X0wsc5G2tuNWPyvhm66JLt0vS1Il1JNQGfok9ZpwoRt631JwhrKQNTouDINrdH6F9CvE=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{e^{\\frac{1}{x}}}{x^{2}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }