derivative of f(x)=5sqrt(x)e^{x^2-3}

{ "query": { "display": "derivative of $$f\\left(x\\right)=5\\sqrt{x}e^{x^{2}-3}$$", "symbolab_question": "PRE_CALC#derivative f(x)=5\\sqrt{x}e^{x^{2}-3}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "5(\\frac{e^{x^{2}-3}}{2\\sqrt{x}}+2e^{x^{2}-3}x\\sqrt{x})", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5\\sqrt{x}e^{x^{2}-3}\\right)=5\\left(\\frac{e^{x^{2}-3}}{2\\sqrt{x}}+2e^{x^{2}-3}x\\sqrt{x}\\right)$$", "input": "\\frac{d}{dx}\\left(5\\sqrt{x}e^{x^{2}-3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=5\\frac{d}{dx}\\left(\\sqrt{x}e^{x^{2}-3}\\right)" }, { "type": "step", "primary": "Apply the Product Rule: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$", "secondary": [ "$$f=\\sqrt{x},\\:g=e^{x^{2}-3}$$" ], "result": "=5\\left(\\frac{d}{dx}\\left(\\sqrt{x}\\right)e^{x^{2}-3}+\\frac{d}{dx}\\left(e^{x^{2}-3}\\right)\\sqrt{x}\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule", "practiceTopic": "Product Rule" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\sqrt{x}\\right)=\\frac{1}{2\\sqrt{x}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{dx}\\left(x^{\\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}x^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}x^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{x}}$$", "input": "\\frac{1}{2}x^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{x}}", "steps": [ { "type": "interim", "title": "$$x^{\\frac{1}{2}-1}=x^{-\\frac{1}{2}}$$", "input": "x^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=x^{-\\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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qijEBDcyPMwV4Y1jeiGyoO0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjQYCY06ctBCI/puUxKEtzAQH2kDe5DGYTz3TrPquGdIjtHZXPNLHlLyai31n5HH4G6M8osviUPEkWv33aMbZrSFQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}x^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$x^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{x}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{x}}", "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{x}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79noABpxc4IZFb3O0CFaPAbH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJrT+HVv/GE/F+xidhei/cy8B9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2ru/qNjapxCbBfMYIYTudnDYTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{x^{2}-3}\\right)=e^{x^{2}-3}\\cdot\\:2x$$", "input": "\\frac{d}{dx}\\left(e^{x^{2}-3}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{x^{2}-3}\\frac{d}{dx}\\left(x^{2}-3\\right)$$", "input": "\\frac{d}{dx}\\left(e^{x^{2}-3}\\right)", "result": "=e^{x^{2}-3}\\frac{d}{dx}\\left(x^{2}-3\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=x^{2}-3$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(x^{2}-3\\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(x^{2}-3\\right)" }, { "type": "step", "primary": "Substitute back $$u=x^{2}-3$$", "result": "=e^{x^{2}-3}\\frac{d}{dx}\\left(x^{2}-3\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtEScpZXYhfWKLYA8boZzOzQMLuvy4VYPXl5uWRTKC4idLl7DeVd7l7l/uUT/v1GhNJLZOXSTCXkMFKW90A2Pi9hxZ2DhIzBSly+pFHZMiI/jR9Dm9WkVXHnLhAMLO7YeI7nOBQF7h3P9btmuDqyvP7WwPs1+Gw97t4MeuaNjSYTyjkVra0ajChSguMMf9fGqJUWlZ+bnpTem3dqAxhtNKM=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}-3\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}-3\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)-\\frac{d}{dx}\\left(3\\right)" }, { "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": "interim", "title": "$$\\frac{d}{dx}\\left(3\\right)=0$$", "input": "\\frac{d}{dx}\\left(3\\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+idP5vLVrjUll6eMdYu6nPER/cBcxgb/Kz63vQV1J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTuwXg0Wd+I5tymlezl5JoPF" } }, { "type": "step", "result": "=2x-0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=e^{x^{2}-3}\\cdot\\:2x" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=5\\left(\\frac{1}{2\\sqrt{x}}e^{x^{2}-3}+e^{x^{2}-3}\\cdot\\:2x\\sqrt{x}\\right)" }, { "type": "interim", "title": "$$\\frac{1}{2\\sqrt{x}}e^{x^{2}-3}=\\frac{e^{x^{2}-3}}{2\\sqrt{x}}$$", "input": "\\frac{1}{2\\sqrt{x}}e^{x^{2}-3}", "result": "=5\\left(\\frac{e^{x^{2}-3}}{2\\sqrt{x}}+2e^{x^{2}-3}x\\sqrt{x}\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:e^{x^{2}-3}}{2\\sqrt{x}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{x^{2}-3}=e^{x^{2}-3}$$", "result": "=\\frac{e^{x^{2}-3}}{2\\sqrt{x}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13CwshGrgmEDbfT9PgpVnbRVPk9CU8MCWpvAxSxvbI1SlD/aL2Coo0GISQwm8bv5wDieB41ExpJobVXcTeiWiZAI88VIRnDSceFlC2THR6cARUo3oe/oyhMy2+1TQhDBd2f4Nsx80xOGoE9kj+ywXXjFSxycNtcA4p0qctwXAzzm574HjUTGkmhtVdxN6JaJkAj0gbeRtkqv7blPmBRbBef9Ykt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice", "practiceTopic": "Derivatives" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=5(\\frac{e^{x^{2}-3}}{2\\sqrt{x}}+2e^{x^{2}-3}x\\sqrt{x})" }, "showViewLarger": true } }, "meta": { "showVerify": true } }