derivative of y=(7ln(x))/(3x+4)

{ "query": { "display": "derivative of $$y=\\frac{7\\ln\\left(x\\right)}{3x+4}$$", "symbolab_question": "PRE_CALC#derivative y=\\frac{7\\ln(x)}{3x+4}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivatives", "subTopic": "Derivatives", "default": "\\frac{7(3x+4-3x\\ln(x))}{x(3x+4)^{2}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{7\\ln\\left(x\\right)}{3x+4}\\right)=\\frac{7\\left(3x+4-3x\\ln\\left(x\\right)\\right)}{x\\left(3x+4\\right)^{2}}$$", "input": "\\frac{d}{dx}\\left(\\frac{7\\ln\\left(x\\right)}{3x+4}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=7\\frac{d}{dx}\\left(\\frac{\\ln\\left(x\\right)}{3x+4}\\right)" }, { "type": "step", "primary": "Apply the Quotient Rule: $$\\left(\\frac{f}{g}\\right)'=\\frac{f'{\\cdot}g-g'{\\cdot}f}{g^{2}}$$", "result": "=7\\cdot\\:\\frac{\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)\\left(3x+4\\right)-\\frac{d}{dx}\\left(3x+4\\right)\\ln\\left(x\\right)}{\\left(3x+4\\right)^{2}}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "result": "=\\frac{1}{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhHxrkiFdmQgNsZN21633mEcjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJlc0OBMs8qTL4oWnxx62vyRTW26qciuyUBGXQExCUedYi3kiAkvXOTkrmcfV8WHLnF4CmnHjYZyazvJkuCAZs64=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(3x+4\\right)=3$$", "input": "\\frac{d}{dx}\\left(3x+4\\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(4\\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": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\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+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "type": "step", "result": "=3+0" }, { "type": "step", "primary": "Simplify", "result": "=3", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=7\\cdot\\:\\frac{\\frac{1}{x}\\left(3x+4\\right)-3\\ln\\left(x\\right)}{\\left(3x+4\\right)^{2}}" }, { "type": "interim", "title": "Simplify $$7\\cdot\\:\\frac{\\frac{1}{x}\\left(3x+4\\right)-3\\ln\\left(x\\right)}{\\left(3x+4\\right)^{2}}:{\\quad}\\frac{7\\left(3x+4-3x\\ln\\left(x\\right)\\right)}{x\\left(3x+4\\right)^{2}}$$", "input": "7\\cdot\\:\\frac{\\frac{1}{x}\\left(3x+4\\right)-3\\ln\\left(x\\right)}{\\left(3x+4\\right)^{2}}", "result": "=\\frac{7\\left(3x+4-3x\\ln\\left(x\\right)\\right)}{x\\left(3x+4\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{1}{x}\\left(3x+4\\right)-3\\ln\\left(x\\right)}{\\left(3x+4\\right)^{2}}=\\frac{3x+4-3x\\ln\\left(x\\right)}{x\\left(3x+4\\right)^{2}}$$", "input": "\\frac{\\frac{1}{x}\\left(3x+4\\right)-3\\ln\\left(x\\right)}{\\left(3x+4\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{x}\\left(3x+4\\right)=\\frac{3x+4}{x}$$", "input": "\\frac{1}{x}\\left(3x+4\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\left(3x+4\\right)}{x}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(3x+4\\right)=3x+4$$", "input": "1\\cdot\\:\\left(3x+4\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(3x+4\\right)=\\left(3x+4\\right)$$", "result": "=\\left(3x+4\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=3x+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77gyRqw1sVDUMwB699iIIpgCWKUbvV6WK3fDUgFtg3Q+WQ/+wFun8kI+v1ohjSjp9IvsJv/VpmBmOllN+wMnDXRbMp+ucL/NNvR56mGR58KSo1u792wvL4Q54HPDS/OY+" } }, { "type": "step", "result": "=\\frac{3x+4}{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MKkUdkEsyxF0d7l2E0e8Xfj+BuNBvhFYqkqGdBMjytXMwViaLUXkeD+JukROhWdjNVfrTb5th/YIkpZ0MsMdQz/L0MoYg+CUn6oyL3EO7YpLkx2OTaWRbYwlV2Ufy4eI03Rw7S3QxQG+3JUiEZJKl2vbp22yN0+LgmMk2HAaaus=" } }, { "type": "step", "result": "=\\frac{\\frac{3x+4}{x}-3\\ln\\left(x\\right)}{\\left(3x+4\\right)^{2}}" }, { "type": "interim", "title": "Join $$\\frac{3x+4}{x}-3\\ln\\left(x\\right):{\\quad}\\frac{3x+4-3x\\ln\\left(x\\right)}{x}$$", "input": "\\frac{3x+4}{x}-3\\ln\\left(x\\right)", "result": "=\\frac{\\frac{3x+4-3x\\ln\\left(x\\right)}{x}}{\\left(3x+4\\right)^{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3\\ln\\left(x\\right)=\\frac{3\\ln\\left(x\\right)x}{x}$$", "result": "=\\frac{3x+4}{x}-\\frac{3\\ln\\left(x\\right)x}{x}" }, { "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{3x+4-3\\ln\\left(x\\right)x}{x}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3x+4-3\\ln\\left(x\\right)x}{x\\left(3x+4\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajTN7bgjh8wX8vuoG/TUqxn+Bv2dUEYzB+KMqoCUj1rStfJEe4bDv60zkwRNMOPq9UULJUwysTkf+ifmThr1HGrpzE/3MJd8ORihzEcyCpF4AjueGMf/3CQxvRiKbM4pYVf8//6/nV5O4fb8Xgwi7maqIWdRsZtwhLu5FToUqQm4FONWwET6j7692QIFXqNYlCIVF9VXMu8rZmMP98ChvH8gs4yze7DEfa/de3ZijbdtwkwTxO72iR5A7D0huGS4VGViVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=7\\cdot\\:\\frac{3x-3x\\ln\\left(x\\right)+4}{x\\left(3x+4\\right)^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(3x+4-3\\ln\\left(x\\right)x\\right)\\cdot\\:7}{x\\left(3x+4\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": 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