(\partial)/(\partial v)(u^2-v^2)

{ "query": { "display": "$$\\frac{\\partial\\:}{\\partial\\:v}\\left(u^{2}-v^{2}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{\\partial }{\\partial v}(u^{2}-v^{2})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Partial Derivatives", "default": "-2v", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:v}\\left(u^{2}-v^{2}\\right)=-2v$$", "input": "\\frac{\\partial\\:}{\\partial\\:v}\\left(u^{2}-v^{2}\\right)", "steps": [ { "type": "step", "primary": "Treat $$u\\:$$as a constant" }, { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{\\partial\\:}{\\partial\\:v}\\left(u^{2}\\right)-\\frac{\\partial\\:}{\\partial\\:v}\\left(v^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:v}\\left(u^{2}\\right)=0$$", "input": "\\frac{\\partial\\:}{\\partial\\:v}\\left(u^{2}\\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+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAmQwnaIErWTuXxWKhJGqPd0HI5S0StY1FdtOqqOPr0TewZLuDut1hvgJ5ZB48DbyVWjeh7+jKEzLb7VNCEMF3Z/ca9U3ttcLbKPPWP74hSSwID8eeMFZbGfCjYpm2CB2mx6ZQejymexKwHa0BQlABl3" } }, { "type": "interim", "title": "$$\\frac{\\partial\\:}{\\partial\\:v}\\left(v^{2}\\right)=2v$$", "input": "\\frac{\\partial\\:}{\\partial\\:v}\\left(v^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2v^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYp9ApVx6pk97jrrFSzJxOAma5CZd2kMrSh7QAScJ03Y1HI5S0StY1FdtOqqOPr0Te00OEdCsXPJu27koksMLDaNkS3dlcCKpQTQcheuut7Mkm+hmRJA1ZPgdMDAPJn089tj8Z+s6gFRao2i0S65lP0EtEYDN3DkVoCoxkVZ3T1C8" } }, { "type": "step", "result": "=0-2v" }, { "type": "step", "primary": "Simplify", "result": "=-2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Partial%20Derivatives", "practiceTopic": "Partial Derivatives" } }, "meta": { "showVerify": true } }