miniaufgabe.js
==== 26. Februar 2024 bis 1. März 2024 ====
=== Dienstag 27. Februar 2024 ===
Leiten Sie von Hand und ohne Unterlagen ab:
miniAufgabe("#exokettenregelvonpoly","#solkettenregelvonpoly",
[["a) $f(x)=\\ln\\left(\\frac{1}{2}x^{3}-\\frac{2}{7}x^{2}-\\frac{3}{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{2}{9}x^{4}-\\frac{1}{7}x+\\frac{3}{5}\\right)}\\quad$ c) $f(x)=\\sqrt{-\\frac{1}{5}x^{3}-\\frac{1}{2}x^{2}-\\frac{2}{5}x}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{1}{2}x^{3}-\\frac{2}{7}x^{2}-\\frac{3}{2}}\\cdot \\left(\\frac{3}{2}x^{2}-\\frac{4}{7}x\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{2}{9}x^{4}-\\frac{1}{7}x+\\frac{3}{5}\\right)}\\cdot \\left(\\frac{8}{9}x^{3}-\\frac{1}{7}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{-\\frac{1}{5}x^{3}-\\frac{1}{2}x^{2}-\\frac{2}{5}x}}\\cdot \\left(-\\frac{3}{5}x^{2}-x-\\frac{2}{5}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{4}{9}x^{3}-\\frac{4}{9}x+\\frac{1}{5}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{1}{8}x^{2}+\\frac{1}{4}x-\\frac{2}{7}\\right)}\\quad$ c) $f(x)=\\sqrt{-\\frac{3}{8}x^{3}-\\frac{2}{7}x-\\frac{4}{9}}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{4}{9}x^{3}-\\frac{4}{9}x+\\frac{1}{5}}\\cdot \\left(-\\frac{4}{3}x^{2}-\\frac{4}{9}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{1}{8}x^{2}+\\frac{1}{4}x-\\frac{2}{7}\\right)}\\cdot \\left(\\frac{1}{4}x+\\frac{1}{4}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{-\\frac{3}{8}x^{3}-\\frac{2}{7}x-\\frac{4}{9}}}\\cdot \\left(-\\frac{9}{8}x^{2}-\\frac{2}{7}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{1}{4}x^{4}-\\frac{1}{7}x^{3}-\\frac{1}{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{2}{7}x^{4}-\\frac{1}{9}x^{3}+\\frac{1}{3}x^{2}\\right)}\\quad$ c) $f(x)=\\sqrt{\\frac{4}{3}x^{3}-\\frac{1}{4}x+\\frac{1}{2}}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{1}{4}x^{4}-\\frac{1}{7}x^{3}-\\frac{1}{2}}\\cdot \\left(x^{3}-\\frac{3}{7}x^{2}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{2}{7}x^{4}-\\frac{1}{9}x^{3}+\\frac{1}{3}x^{2}\\right)}\\cdot \\left(\\frac{8}{7}x^{3}-\\frac{1}{3}x^{2}+\\frac{2}{3}x\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{\\frac{4}{3}x^{3}-\\frac{1}{4}x+\\frac{1}{2}}}\\cdot \\left(4x^{2}-\\frac{1}{4}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{1}{7}x^{2}-\\frac{2}{7}x-\\frac{1}{8}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{1}{2}x^{4}-\\frac{4}{5}x^{3}+\\frac{2}{3}\\right)}\\quad$ c) $f(x)=\\sqrt{\\frac{1}{3}x^{3}-\\frac{2}{3}x^{2}-\\frac{4}{3}x}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{1}{7}x^{2}-\\frac{2}{7}x-\\frac{1}{8}}\\cdot \\left(-\\frac{2}{7}x-\\frac{2}{7}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{1}{2}x^{4}-\\frac{4}{5}x^{3}+\\frac{2}{3}\\right)}\\cdot \\left(2x^{3}-\\frac{12}{5}x^{2}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{\\frac{1}{3}x^{3}-\\frac{2}{3}x^{2}-\\frac{4}{3}x}}\\cdot \\left(x^{2}-\\frac{4}{3}x-\\frac{4}{3}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{1}{2}x^{4}+\\frac{1}{2}x+\\frac{2}{5}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{3}{8}x^{3}+\\frac{2}{7}x^{2}+\\frac{4}{7}x\\right)}\\quad$ c) $f(x)=\\sqrt{\\frac{4}{9}x^{3}-\\frac{1}{8}x^{2}+\\frac{1}{5}x}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{1}{2}x^{4}+\\frac{1}{2}x+\\frac{2}{5}}\\cdot \\left(2x^{3}+\\frac{1}{2}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{3}{8}x^{3}+\\frac{2}{7}x^{2}+\\frac{4}{7}x\\right)}\\cdot \\left(-\\frac{9}{8}x^{2}+\\frac{4}{7}x+\\frac{4}{7}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{\\frac{4}{9}x^{3}-\\frac{1}{8}x^{2}+\\frac{1}{5}x}}\\cdot \\left(\\frac{4}{3}x^{2}-\\frac{1}{4}x+\\frac{1}{5}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{1}{7}x^{4}-\\frac{2}{3}x^{2}-\\frac{1}{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{1}{8}x^{4}-\\frac{1}{2}x^{3}-\\frac{1}{4}x\\right)}\\quad$ c) $f(x)=\\sqrt{-\\frac{1}{4}x^{4}-\\frac{2}{5}x^{3}+\\frac{1}{8}x^{2}}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{1}{7}x^{4}-\\frac{2}{3}x^{2}-\\frac{1}{2}}\\cdot \\left(\\frac{4}{7}x^{3}-\\frac{4}{3}x\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{1}{8}x^{4}-\\frac{1}{2}x^{3}-\\frac{1}{4}x\\right)}\\cdot \\left(-\\frac{1}{2}x^{3}-\\frac{3}{2}x^{2}-\\frac{1}{4}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{-\\frac{1}{4}x^{4}-\\frac{2}{5}x^{3}+\\frac{1}{8}x^{2}}}\\cdot \\left(-x^{3}-\\frac{6}{5}x^{2}+\\frac{1}{4}x\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{1}{3}x^{3}-\\frac{1}{2}x+\\frac{2}{5}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{3}{4}x^{4}+\\frac{1}{2}x^{3}+\\frac{4}{5}x^{2}\\right)}\\quad$ c) $f(x)=\\sqrt{\\frac{2}{9}x^{4}+\\frac{4}{7}x^{2}-\\frac{4}{3}x}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{1}{3}x^{3}-\\frac{1}{2}x+\\frac{2}{5}}\\cdot \\left(-x^{2}-\\frac{1}{2}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{3}{4}x^{4}+\\frac{1}{2}x^{3}+\\frac{4}{5}x^{2}\\right)}\\cdot \\left(-3x^{3}+\\frac{3}{2}x^{2}+\\frac{8}{5}x\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{\\frac{2}{9}x^{4}+\\frac{4}{7}x^{2}-\\frac{4}{3}x}}\\cdot \\left(\\frac{8}{9}x^{3}+\\frac{8}{7}x-\\frac{4}{3}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{1}{9}x^{4}-\\frac{3}{7}x^{3}-\\frac{4}{3}x^{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{2}{9}x^{4}+\\frac{1}{2}x^{2}-\\frac{2}{3}x\\right)}\\quad$ c) $f(x)=\\sqrt{-\\frac{4}{7}x^{4}-\\frac{2}{9}x^{3}+\\frac{4}{9}x}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{1}{9}x^{4}-\\frac{3}{7}x^{3}-\\frac{4}{3}x^{2}}\\cdot \\left(\\frac{4}{9}x^{3}-\\frac{9}{7}x^{2}-\\frac{8}{3}x\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{2}{9}x^{4}+\\frac{1}{2}x^{2}-\\frac{2}{3}x\\right)}\\cdot \\left(\\frac{8}{9}x^{3}+x-\\frac{2}{3}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{-\\frac{4}{7}x^{4}-\\frac{2}{9}x^{3}+\\frac{4}{9}x}}\\cdot \\left(-\\frac{16}{7}x^{3}-\\frac{2}{3}x^{2}+\\frac{4}{9}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{1}{2}x^{4}-\\frac{4}{9}x^{2}-\\frac{2}{3}x\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{1}{3}x^{3}+\\frac{2}{3}x-\\frac{2}{3}\\right)}\\quad$ c) $f(x)=\\sqrt{-\\frac{1}{2}x^{3}+\\frac{1}{4}x-\\frac{1}{4}}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{1}{2}x^{4}-\\frac{4}{9}x^{2}-\\frac{2}{3}x}\\cdot \\left(-2x^{3}-\\frac{8}{9}x-\\frac{2}{3}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{1}{3}x^{3}+\\frac{2}{3}x-\\frac{2}{3}\\right)}\\cdot \\left(-x^{2}+\\frac{2}{3}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{-\\frac{1}{2}x^{3}+\\frac{1}{4}x-\\frac{1}{4}}}\\cdot \\left(-\\frac{3}{2}x^{2}+\\frac{1}{4}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{3}{2}x^{4}+\\frac{3}{7}x^{2}-\\frac{1}{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{3}{4}x^{4}-\\frac{1}{8}x+\\frac{1}{3}\\right)}\\quad$ c) $f(x)=\\sqrt{\\frac{1}{6}x^{3}-\\frac{2}{7}x^{2}-\\frac{2}{3}x}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{3}{2}x^{4}+\\frac{3}{7}x^{2}-\\frac{1}{2}}\\cdot \\left(6x^{3}+\\frac{6}{7}x\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{3}{4}x^{4}-\\frac{1}{8}x+\\frac{1}{3}\\right)}\\cdot \\left(-3x^{3}-\\frac{1}{8}\\right)\\quad$ c) $f'(x)=\\frac{1}{2\\sqrt{\\frac{1}{6}x^{3}-\\frac{2}{7}x^{2}-\\frac{2}{3}x}}\\cdot \\left(\\frac{1}{2}x^{2}-\\frac{4}{7}x-\\frac{2}{3}\\right)\\quad$ "]],
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ruby ableiten-von-hand.rb 2
=== Mittwoch 28. Februar 2024 ====
Leiten Sie von Hand und ohne Unterlagen ab. Geben Sie das Resultat in der Form "Polynom + $\ln(x)$ mal Polynom" an.
miniAufgabe("#exoprodlnpoly","#solprodlnpoly",
[["$\\frac{4}{3} \\cdot \\ln(x) \\cdot \\left(\\frac{1}{2}x^{3}-\\frac{4}{7}x\\right)$", "$\\frac{4}{3} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(\\frac{1}{2}x^{3}-\\frac{4}{7}x\\right) + \\ln(x) \\cdot \\left(\\frac{3}{2}x^{2}-\\frac{4}{7}\\right)\\Biggr) = \\frac{4}{3} \\cdot \\left(\\frac{1}{2}x^{2}-\\frac{4}{7}\\right) \\quad+\\frac{4}{3} \\cdot \\ln(x) \\cdot \\left(\\frac{3}{2}x^{2}-\\frac{4}{7}\\right) = \\frac{2}{3}x^{2}-\\frac{16}{21} + \\ln(x) \\cdot \\left(2x^{2}-\\frac{16}{21}\\right)$"], ["$\\frac{3}{5} \\cdot \\ln(x) \\cdot \\left(\\frac{3}{5}x^{3}-\\frac{3}{8}\\right)$", "$\\frac{3}{5} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(\\frac{3}{5}x^{3}-\\frac{3}{8}\\right) + \\ln(x) \\cdot \\left(\\frac{9}{5}x^{2}\\right)\\Biggr) = \\frac{3}{5} \\cdot \\left(\\frac{3}{5}x^{2}-\\frac{3}{8} \\cdot \\frac{1}{x}\\right) \\quad+\\frac{3}{5} \\cdot \\ln(x) \\cdot \\left(\\frac{9}{5}x^{2}\\right) = \\frac{9}{25}x^{2}-\\frac{9}{40} \\cdot \\frac{1}{x} + \\ln(x) \\cdot \\left(\\frac{27}{25}x^{2}\\right)$"], ["$-\\frac{1}{2} \\cdot \\ln(x) \\cdot \\left(-\\frac{2}{9}x^{3}+\\frac{1}{2}\\right)$", "$-\\frac{1}{2} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(-\\frac{2}{9}x^{3}+\\frac{1}{2}\\right) + \\ln(x) \\cdot \\left(-\\frac{2}{3}x^{2}\\right)\\Biggr) = -\\frac{1}{2} \\cdot \\left(-\\frac{2}{9}x^{2}+\\frac{1}{2} \\cdot \\frac{1}{x}\\right) \\quad-\\frac{1}{2} \\cdot \\ln(x) \\cdot \\left(-\\frac{2}{3}x^{2}\\right) = \\frac{1}{9}x^{2}-\\frac{1}{4} \\cdot \\frac{1}{x} + \\ln(x) \\cdot \\left(\\frac{1}{3}x^{2}\\right)$"], ["$\\frac{4}{9} \\cdot \\ln(x) \\cdot \\left(-\\frac{1}{2}x^{4}+\\frac{1}{8}x\\right)$", "$\\frac{4}{9} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(-\\frac{1}{2}x^{4}+\\frac{1}{8}x\\right) + \\ln(x) \\cdot \\left(-2x^{3}+\\frac{1}{8}\\right)\\Biggr) = \\frac{4}{9} \\cdot \\left(-\\frac{1}{2}x^{3}+\\frac{1}{8}\\right) \\quad+\\frac{4}{9} \\cdot \\ln(x) \\cdot \\left(-2x^{3}+\\frac{1}{8}\\right) = -\\frac{2}{9}x^{3}+\\frac{1}{18} + \\ln(x) \\cdot \\left(-\\frac{8}{9}x^{3}+\\frac{1}{18}\\right)$"], ["$\\frac{3}{5} \\cdot \\ln(x) \\cdot \\left(-\\frac{1}{7}x^{3}+\\frac{1}{2}\\right)$", "$\\frac{3}{5} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(-\\frac{1}{7}x^{3}+\\frac{1}{2}\\right) + \\ln(x) \\cdot \\left(-\\frac{3}{7}x^{2}\\right)\\Biggr) = \\frac{3}{5} \\cdot \\left(-\\frac{1}{7}x^{2}+\\frac{1}{2} \\cdot \\frac{1}{x}\\right) \\quad+\\frac{3}{5} \\cdot \\ln(x) \\cdot \\left(-\\frac{3}{7}x^{2}\\right) = -\\frac{3}{35}x^{2}+\\frac{3}{10} \\cdot \\frac{1}{x} + \\ln(x) \\cdot \\left(-\\frac{9}{35}x^{2}\\right)$"], ["$-\\frac{1}{4} \\cdot \\ln(x) \\cdot \\left(-\\frac{1}{2}x^{4}-\\frac{1}{8}x^{3}\\right)$", "$-\\frac{1}{4} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(-\\frac{1}{2}x^{4}-\\frac{1}{8}x^{3}\\right) + \\ln(x) \\cdot \\left(-2x^{3}-\\frac{3}{8}x^{2}\\right)\\Biggr) = -\\frac{1}{4} \\cdot \\left(-\\frac{1}{2}x^{3}-\\frac{1}{8}x^{2}\\right) \\quad-\\frac{1}{4} \\cdot \\ln(x) \\cdot \\left(-2x^{3}-\\frac{3}{8}x^{2}\\right) = \\frac{1}{8}x^{3}+\\frac{1}{32}x^{2} + \\ln(x) \\cdot \\left(\\frac{1}{2}x^{3}+\\frac{3}{32}x^{2}\\right)$"], ["$-\\frac{4}{9} \\cdot \\ln(x) \\cdot \\left(\\frac{1}{9}x-\\frac{2}{9}\\right)$", "$-\\frac{4}{9} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(\\frac{1}{9}x-\\frac{2}{9}\\right) + \\ln(x) \\cdot \\left(\\frac{1}{9}\\right)\\Biggr) = -\\frac{4}{9} \\cdot \\left(\\frac{1}{9}-\\frac{2}{9} \\cdot \\frac{1}{x}\\right) \\quad-\\frac{4}{9} \\cdot \\ln(x) \\cdot \\left(\\frac{1}{9}\\right) = -\\frac{4}{81}+\\frac{8}{81} \\cdot \\frac{1}{x} + \\ln(x) \\cdot \\left(-\\frac{4}{81}\\right)$"], ["$\\frac{1}{4} \\cdot \\ln(x) \\cdot \\left(-\\frac{2}{9}x^{2}-\\frac{4}{5}\\right)$", "$\\frac{1}{4} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(-\\frac{2}{9}x^{2}-\\frac{4}{5}\\right) + \\ln(x) \\cdot \\left(-\\frac{4}{9}x\\right)\\Biggr) = \\frac{1}{4} \\cdot \\left(-\\frac{2}{9}x-\\frac{4}{5} \\cdot \\frac{1}{x}\\right) \\quad+\\frac{1}{4} \\cdot \\ln(x) \\cdot \\left(-\\frac{4}{9}x\\right) = -\\frac{1}{18}x-\\frac{1}{5} \\cdot \\frac{1}{x} + \\ln(x) \\cdot \\left(-\\frac{1}{9}x\\right)$"], ["$-\\frac{4}{5} \\cdot \\ln(x) \\cdot \\left(\\frac{1}{9}x^{4}-\\frac{4}{7}x\\right)$", "$-\\frac{4}{5} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(\\frac{1}{9}x^{4}-\\frac{4}{7}x\\right) + \\ln(x) \\cdot \\left(\\frac{4}{9}x^{3}-\\frac{4}{7}\\right)\\Biggr) = -\\frac{4}{5} \\cdot \\left(\\frac{1}{9}x^{3}-\\frac{4}{7}\\right) \\quad-\\frac{4}{5} \\cdot \\ln(x) \\cdot \\left(\\frac{4}{9}x^{3}-\\frac{4}{7}\\right) = -\\frac{4}{45}x^{3}+\\frac{16}{35} + \\ln(x) \\cdot \\left(-\\frac{16}{45}x^{3}+\\frac{16}{35}\\right)$"], ["$-\\frac{4}{3} \\cdot \\ln(x) \\cdot \\left(\\frac{1}{3}x+\\frac{1}{9}\\right)$", "$-\\frac{4}{3} \\cdot \\Biggl( \\frac{1}{x} \\cdot \\left(\\frac{1}{3}x+\\frac{1}{9}\\right) + \\ln(x) \\cdot \\left(\\frac{1}{3}\\right)\\Biggr) = -\\frac{4}{3} \\cdot \\left(\\frac{1}{3}+\\frac{1}{9} \\cdot \\frac{1}{x}\\right) \\quad-\\frac{4}{3} \\cdot \\ln(x) \\cdot \\left(\\frac{1}{3}\\right) = -\\frac{4}{9}-\\frac{4}{27} \\cdot \\frac{1}{x} + \\ln(x) \\cdot \\left(-\\frac{4}{9}\\right)$"]],
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ruby ableiten-von-hand.rb 6