derivative of (3x+2/(x-3))

{ "query": { "display": "$$\\frac{d}{dx}\\left(\\frac{3x+2}{x-3}\\right)$$", "symbolab_question": "DERIVATIVE#\\frac{d}{dx}(\\frac{3x+2}{x-3})" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "-\\frac{11}{(x-3)^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{3x+2}{x-3}\\right)=-\\frac{11}{\\left(x-3\\right)^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{3x+2}{x-3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=\\frac{\\frac{d}{dx}\\left(3x+2\\right)\\left(x-3\\right)-\\frac{d}{dx}\\left(x-3\\right)\\left(3x+2\\right)}{\\left(x-3\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x+2\\right)=3$$", "input": "\\frac{d}{dx}\\left(3x+2\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(3x\\right)+\\frac{d}{dx}\\left(2\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x\\right)=3$$", "input": "\\frac{d}{dx}\\left(3x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=3\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=3\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsUnAVaDXCLQOuynYB+k3fTZGku9zFkxwe1dTH8vycb9BbqPJ4kl+ElAajU+EBTcW1NbbqpyK7JQEZdATEJR51i3CwF9WgU8/+rFW242YnYD" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2\\right)=0$$", "input": "\\frac{d}{dx}\\left(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+idP5vLVrjUll6eMdYiiraNd5UTAiEFXslV0UVyVJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtRm0l+ci6m9OnlYfI6EjHe" } }, { "type": "step", "result": "=3+0" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x-3\\right)=1$$", "input": "\\frac{d}{dx}\\left(x-3\\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(3\\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": "$$\\frac{d}{dx}\\left(3\\right)=0$$", "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": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{3\\left(x-3\\right)-1\\cdot\\:\\left(3x+2\\right)}{\\left(x-3\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{3\\left(x-3\\right)-1\\cdot\\:\\left(3x+2\\right)}{\\left(x-3\\right)^{2}}:{\\quad}-\\frac{11}{\\left(x-3\\right)^{2}}$$", "input": "\\frac{3\\left(x-3\\right)-1\\cdot\\:\\left(3x+2\\right)}{\\left(x-3\\right)^{2}}", "result": "=-\\frac{11}{\\left(x-3\\right)^{2}}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(3x+2\\right)=\\left(3x+2\\right)$$", "result": "=\\frac{3\\left(x-3\\right)-\\left(3x+2\\right)}{\\left(x-3\\right)^{2}}" }, { "type": "interim", "title": "Expand $$3\\left(x-3\\right)-\\left(3x+2\\right):{\\quad}-11$$", "input": "3\\left(x-3\\right)-\\left(3x+2\\right)", "result": "=\\frac{-11}{\\left(x-3\\right)^{2}}", "steps": [ { "type": "interim", "title": "Expand $$3\\left(x-3\\right):{\\quad}3x-9$$", "input": "3\\left(x-3\\right)", "result": "=3x-9-\\left(3x+2\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=3,\\:b=x,\\:c=3$$" ], "result": "=3x-3\\cdot\\:3", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=3x-9" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76bEBDa/xY7DOP+55X7h8LHWD310L1+P2yDQQfMEhENGUU2TCbVhirFAvEIzHoNPr1sD7NfhsPe7eDHrmjY0mEzKnzQxYP7MiHWJB4erexo+wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$-\\left(3x+2\\right):{\\quad}-3x-2$$", "input": "-\\left(3x+2\\right)", "result": "=3x-9-3x-2", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(3x\\right)-\\left(2\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-3x-2" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Simplify $$3x-9-3x-2:{\\quad}-11$$", "input": "3x-9-3x-2", "result": "=-11", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=3x-3x-9-2" }, { "type": "step", "primary": "Add similar elements: $$3x-3x=0$$", "result": "=-9-2" }, { "type": "step", "primary": "Subtract the numbers: $$-9-2=-11$$", "result": "=-11" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78rr4B/AoBtx0zBrCa1DDpC061ljBSPJeENOw2efoSWszOI7WO7uWw0n4IkiIixhBZEt3ZXAiqUE0HIXrrrezJFTofSdL77yQOo/rfZ+5Uc83/eF74dgynaxu/P+ewKYl" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{11}{\\left(x-3\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s772FiZpcrxP5G+eEBqGkN7bgwCmb21/8fCl90ciQOwIHqXLU+bqNuXJ7miYz5zxIEzRqDxPUzBN6vjj5oJL9kUK0SjENKQPAOu45ZmzDJARdunKQHR8b6odod76H2AT4uRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6opzhIyw9oLG7jv/LxFxJXtaGdG+HCb8u2VMw8GZNTIrlfdL7xdD/GfAFaPgS97jDM=" } } ], "meta": { "solvingClass": "Derivatives", "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Quotient%20Rule", "practiceTopic": "Quotient Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{11}{(x-3)^{2}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }