miniaufgabe.js ==== 18. Februar 2019 bis 22. Februar 2019 ==== === Mittwoch 20. Februar 2019 === **Reminder** für die 4lW: 10:55 Fototermin Innentreppe Eingang Lämmlisbrunnstrasse. Leiten Sie von Hand und ohne Unterlagen ab: miniAufgabe("#exotrigpoly","#soltrigpoly", [["a) $f(x)=-\\frac{1}{2}\\cdot\\cos\\left(\\frac{4}{9}x^{4}+\\frac{4}{5}x^{2}\\right)\\quad$ b) $f(x)=\\frac{1}{2}\\cdot\\sin\\left(\\frac{1}{3}x^{4}-\\frac{1}{4}x^{2}\\right)\\quad$ ", "a) $f'(x)=-\\frac{1}{2} \\cdot \\left(\\frac{16}{9}x^{3}+\\frac{8}{5}x\\right) \\cdot \\left(-\\sin\\left(\\frac{4}{9}x^{4}+\\frac{4}{5}x^{2}\\right)\\right) = \\left(\\frac{8}{9}x^{3}+\\frac{4}{5}x\\right)\\cdot \\sin\\left(\\frac{4}{9}x^{4}+\\frac{4}{5}x^{2}\\right)$
b) $f'(x)=\\frac{1}{2} \\cdot \\left(\\frac{4}{3}x^{3}-\\frac{1}{2}x\\right) \\cdot \\cos\\left(\\frac{1}{3}x^{4}-\\frac{1}{4}x^{2}\\right) = \\left(\\frac{2}{3}x^{3}-\\frac{1}{4}x\\right)\\cdot \\cos\\left(\\frac{1}{3}x^{4}-\\frac{1}{4}x^{2}\\right)$
"], ["a) $f(x)=\\frac{4}{5}\\cdot\\cos\\left(\\frac{1}{2}x^{4}-\\frac{1}{3}x^{3}\\right)\\quad$ b) $f(x)=\\frac{1}{4}\\cdot\\sin\\left(\\frac{1}{4}x^{2}-\\frac{2}{9}x\\right)\\quad$ ", "a) $f'(x)=\\frac{4}{5} \\cdot \\left(2x^{3}-x^{2}\\right) \\cdot \\left(-\\sin\\left(\\frac{1}{2}x^{4}-\\frac{1}{3}x^{3}\\right)\\right) = \\left(-\\frac{8}{5}x^{3}+\\frac{4}{5}x^{2}\\right)\\cdot \\sin\\left(\\frac{1}{2}x^{4}-\\frac{1}{3}x^{3}\\right)$
b) $f'(x)=\\frac{1}{4} \\cdot \\left(\\frac{1}{2}x-\\frac{2}{9}\\right) \\cdot \\cos\\left(\\frac{1}{4}x^{2}-\\frac{2}{9}x\\right) = \\left(\\frac{1}{8}x-\\frac{1}{18}\\right)\\cdot \\cos\\left(\\frac{1}{4}x^{2}-\\frac{2}{9}x\\right)$
"], ["a) $f(x)=-\\frac{2}{7}\\cdot\\cos\\left(-\\frac{1}{4}x^{3}+\\frac{2}{9}\\right)\\quad$ b) $f(x)=-\\frac{1}{4}\\cdot\\sin\\left(-\\frac{2}{9}x^{3}-\\frac{1}{7}x^{2}\\right)\\quad$ ", "a) $f'(x)=-\\frac{2}{7} \\cdot \\left(-\\frac{3}{4}x^{2}\\right) \\cdot \\left(-\\sin\\left(-\\frac{1}{4}x^{3}+\\frac{2}{9}\\right)\\right) = -\\frac{3}{14}x^{2}\\cdot \\sin\\left(-\\frac{1}{4}x^{3}+\\frac{2}{9}\\right)$
b) $f'(x)=-\\frac{1}{4} \\cdot \\left(-\\frac{2}{3}x^{2}-\\frac{2}{7}x\\right) \\cdot \\cos\\left(-\\frac{2}{9}x^{3}-\\frac{1}{7}x^{2}\\right) = \\left(\\frac{1}{6}x^{2}+\\frac{1}{14}x\\right)\\cdot \\cos\\left(-\\frac{2}{9}x^{3}-\\frac{1}{7}x^{2}\\right)$
"], ["a) $f(x)=\\frac{1}{6}\\cdot\\cos\\left(-\\frac{1}{2}x^{3}-\\frac{1}{2}x\\right)\\quad$ b) $f(x)=-\\frac{1}{3}\\cdot\\sin\\left(\\frac{1}{2}x^{2}+\\frac{1}{6}\\right)\\quad$ ", "a) $f'(x)=\\frac{1}{6} \\cdot \\left(-\\frac{3}{2}x^{2}-\\frac{1}{2}\\right) \\cdot \\left(-\\sin\\left(-\\frac{1}{2}x^{3}-\\frac{1}{2}x\\right)\\right) = \\left(\\frac{1}{4}x^{2}+\\frac{1}{12}\\right)\\cdot \\sin\\left(-\\frac{1}{2}x^{3}-\\frac{1}{2}x\\right)$
b) $f'(x)=-\\frac{1}{3} \\cdot \\left(x\\right) \\cdot \\cos\\left(\\frac{1}{2}x^{2}+\\frac{1}{6}\\right) = -\\frac{1}{3}x\\cdot \\cos\\left(\\frac{1}{2}x^{2}+\\frac{1}{6}\\right)$
"], ["a) $f(x)=-\\frac{1}{5}\\cdot\\cos\\left(-\\frac{1}{6}x^{3}-\\frac{1}{3}x\\right)\\quad$ b) $f(x)=-\\frac{1}{5}\\cdot\\sin\\left(-\\frac{1}{4}x^{4}+\\frac{2}{3}x^{3}\\right)\\quad$ ", "a) $f'(x)=-\\frac{1}{5} \\cdot \\left(-\\frac{1}{2}x^{2}-\\frac{1}{3}\\right) \\cdot \\left(-\\sin\\left(-\\frac{1}{6}x^{3}-\\frac{1}{3}x\\right)\\right) = \\left(-\\frac{1}{10}x^{2}-\\frac{1}{15}\\right)\\cdot \\sin\\left(-\\frac{1}{6}x^{3}-\\frac{1}{3}x\\right)$
b) $f'(x)=-\\frac{1}{5} \\cdot \\left(-x^{3}+2x^{2}\\right) \\cdot \\cos\\left(-\\frac{1}{4}x^{4}+\\frac{2}{3}x^{3}\\right) = \\left(\\frac{1}{5}x^{3}-\\frac{2}{5}x^{2}\\right)\\cdot \\cos\\left(-\\frac{1}{4}x^{4}+\\frac{2}{3}x^{3}\\right)$
"], ["a) $f(x)=\\frac{1}{2}\\cdot\\cos\\left(-\\frac{1}{5}x^{3}-\\frac{1}{4}x\\right)\\quad$ b) $f(x)=-\\frac{1}{9}\\cdot\\sin\\left(-\\frac{4}{7}x^{4}+\\frac{3}{7}x\\right)\\quad$ ", "a) $f'(x)=\\frac{1}{2} \\cdot \\left(-\\frac{3}{5}x^{2}-\\frac{1}{4}\\right) \\cdot \\left(-\\sin\\left(-\\frac{1}{5}x^{3}-\\frac{1}{4}x\\right)\\right) = \\left(\\frac{3}{10}x^{2}+\\frac{1}{8}\\right)\\cdot \\sin\\left(-\\frac{1}{5}x^{3}-\\frac{1}{4}x\\right)$
b) $f'(x)=-\\frac{1}{9} \\cdot \\left(-\\frac{16}{7}x^{3}+\\frac{3}{7}\\right) \\cdot \\cos\\left(-\\frac{4}{7}x^{4}+\\frac{3}{7}x\\right) = \\left(\\frac{16}{63}x^{3}-\\frac{1}{21}\\right)\\cdot \\cos\\left(-\\frac{4}{7}x^{4}+\\frac{3}{7}x\\right)$
"], ["a) $f(x)=-\\frac{1}{2}\\cdot\\cos\\left(\\frac{3}{4}x^{4}-\\frac{4}{9}x^{2}\\right)\\quad$ b) $f(x)=-\\frac{1}{3}\\cdot\\sin\\left(\\frac{3}{2}x^{2}+\\frac{4}{9}\\right)\\quad$ ", "a) $f'(x)=-\\frac{1}{2} \\cdot \\left(3x^{3}-\\frac{8}{9}x\\right) \\cdot \\left(-\\sin\\left(\\frac{3}{4}x^{4}-\\frac{4}{9}x^{2}\\right)\\right) = \\left(\\frac{3}{2}x^{3}-\\frac{4}{9}x\\right)\\cdot \\sin\\left(\\frac{3}{4}x^{4}-\\frac{4}{9}x^{2}\\right)$
b) $f'(x)=-\\frac{1}{3} \\cdot \\left(3x\\right) \\cdot \\cos\\left(\\frac{3}{2}x^{2}+\\frac{4}{9}\\right) = -x\\cdot \\cos\\left(\\frac{3}{2}x^{2}+\\frac{4}{9}\\right)$
"], ["a) $f(x)=\\frac{2}{5}\\cdot\\cos\\left(-\\frac{2}{3}x^{3}+\\frac{3}{5}x^{2}\\right)\\quad$ b) $f(x)=\\frac{1}{6}\\cdot\\sin\\left(-\\frac{3}{2}x^{4}+\\frac{3}{7}x^{3}\\right)\\quad$ ", "a) $f'(x)=\\frac{2}{5} \\cdot \\left(-2x^{2}+\\frac{6}{5}x\\right) \\cdot \\left(-\\sin\\left(-\\frac{2}{3}x^{3}+\\frac{3}{5}x^{2}\\right)\\right) = \\left(\\frac{4}{5}x^{2}-\\frac{12}{25}x\\right)\\cdot \\sin\\left(-\\frac{2}{3}x^{3}+\\frac{3}{5}x^{2}\\right)$
b) $f'(x)=\\frac{1}{6} \\cdot \\left(-6x^{3}+\\frac{9}{7}x^{2}\\right) \\cdot \\cos\\left(-\\frac{3}{2}x^{4}+\\frac{3}{7}x^{3}\\right) = \\left(-x^{3}+\\frac{3}{14}x^{2}\\right)\\cdot \\cos\\left(-\\frac{3}{2}x^{4}+\\frac{3}{7}x^{3}\\right)$
"], ["a) $f(x)=\\frac{4}{3}\\cdot\\cos\\left(\\frac{1}{3}x^{4}-\\frac{1}{2}\\right)\\quad$ b) $f(x)=-\\frac{3}{8}\\cdot\\sin\\left(-\\frac{1}{2}x^{3}-\\frac{1}{2}\\right)\\quad$ ", "a) $f'(x)=\\frac{4}{3} \\cdot \\left(\\frac{4}{3}x^{3}\\right) \\cdot \\left(-\\sin\\left(\\frac{1}{3}x^{4}-\\frac{1}{2}\\right)\\right) = -\\frac{16}{9}x^{3}\\cdot \\sin\\left(\\frac{1}{3}x^{4}-\\frac{1}{2}\\right)$
b) $f'(x)=-\\frac{3}{8} \\cdot \\left(-\\frac{3}{2}x^{2}\\right) \\cdot \\cos\\left(-\\frac{1}{2}x^{3}-\\frac{1}{2}\\right) = \\frac{9}{16}x^{2}\\cdot \\cos\\left(-\\frac{1}{2}x^{3}-\\frac{1}{2}\\right)$
"], ["a) $f(x)=\\frac{1}{9}\\cdot\\cos\\left(\\frac{1}{8}x^{2}+\\frac{3}{2}x\\right)\\quad$ b) $f(x)=\\frac{3}{4}\\cdot\\sin\\left(-\\frac{1}{3}x^{4}+\\frac{3}{7}x^{2}\\right)\\quad$ ", "a) $f'(x)=\\frac{1}{9} \\cdot \\left(\\frac{1}{4}x+\\frac{3}{2}\\right) \\cdot \\left(-\\sin\\left(\\frac{1}{8}x^{2}+\\frac{3}{2}x\\right)\\right) = \\left(-\\frac{1}{36}x-\\frac{1}{6}\\right)\\cdot \\sin\\left(\\frac{1}{8}x^{2}+\\frac{3}{2}x\\right)$
b) $f'(x)=\\frac{3}{4} \\cdot \\left(-\\frac{4}{3}x^{3}+\\frac{6}{7}x\\right) \\cdot \\cos\\left(-\\frac{1}{3}x^{4}+\\frac{3}{7}x^{2}\\right) = \\left(-x^{3}+\\frac{9}{14}x\\right)\\cdot \\cos\\left(-\\frac{1}{3}x^{4}+\\frac{3}{7}x^{2}\\right)$
"]], "
");

Es geht hier auch um die Kettenregel: $\left(f(g(x))\right)' = f'(g(x)) \cdot g'(x)$

=== Freitag 22. Februar 2019 === Leiten Sie von Hand und ohne Unterlagen ab: miniAufgabe("#exokettenregelvonpoly","#solkettenregelvonpoly", [["a) $f(x)=\\ln\\left(-\\frac{1}{3}x^{2}+\\frac{1}{8}x+\\frac{3}{5}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{1}{7}x^{4}-\\frac{1}{4}x^{3}-\\frac{1}{2}\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{1}{3}x^{2}+\\frac{1}{8}x+\\frac{3}{5}}\\cdot \\left(-\\frac{2}{3}x+\\frac{1}{8}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{1}{7}x^{4}-\\frac{1}{4}x^{3}-\\frac{1}{2}\\right)}\\cdot \\left(-\\frac{4}{7}x^{3}-\\frac{3}{4}x^{2}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{4}{7}x^{3}-\\frac{3}{8}x+\\frac{1}{4}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{2}{5}x^{4}+\\frac{2}{3}x+\\frac{1}{2}\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{4}{7}x^{3}-\\frac{3}{8}x+\\frac{1}{4}}\\cdot \\left(\\frac{12}{7}x^{2}-\\frac{3}{8}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{2}{5}x^{4}+\\frac{2}{3}x+\\frac{1}{2}\\right)}\\cdot \\left(-\\frac{8}{5}x^{3}+\\frac{2}{3}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{1}{5}x^{4}-\\frac{1}{2}x-\\frac{1}{7}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{3}{2}x^{4}-\\frac{1}{7}x^{3}-\\frac{4}{5}x^{2}\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{1}{5}x^{4}-\\frac{1}{2}x-\\frac{1}{7}}\\cdot \\left(\\frac{4}{5}x^{3}-\\frac{1}{2}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{3}{2}x^{4}-\\frac{1}{7}x^{3}-\\frac{4}{5}x^{2}\\right)}\\cdot \\left(-6x^{3}-\\frac{3}{7}x^{2}-\\frac{8}{5}x\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{3}{2}x^{4}+\\frac{1}{2}x^{3}+\\frac{1}{2}x\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{4}{9}x^{3}+\\frac{1}{5}x^{2}-\\frac{2}{5}x\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{3}{2}x^{4}+\\frac{1}{2}x^{3}+\\frac{1}{2}x}\\cdot \\left(-6x^{3}+\\frac{3}{2}x^{2}+\\frac{1}{2}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{4}{9}x^{3}+\\frac{1}{5}x^{2}-\\frac{2}{5}x\\right)}\\cdot \\left(\\frac{4}{3}x^{2}+\\frac{2}{5}x-\\frac{2}{5}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{1}{3}x^{3}+\\frac{1}{3}x+\\frac{4}{7}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{2}{7}x^{4}-\\frac{4}{9}x^{3}+\\frac{1}{3}x\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{1}{3}x^{3}+\\frac{1}{3}x+\\frac{4}{7}}\\cdot \\left(-x^{2}+\\frac{1}{3}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{2}{7}x^{4}-\\frac{4}{9}x^{3}+\\frac{1}{3}x\\right)}\\cdot \\left(\\frac{8}{7}x^{3}-\\frac{4}{3}x^{2}+\\frac{1}{3}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{2}{5}x^{4}+\\frac{1}{3}x^{3}+\\frac{3}{2}x^{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{1}{4}x^{3}-\\frac{1}{3}x^{2}+\\frac{4}{3}x\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{2}{5}x^{4}+\\frac{1}{3}x^{3}+\\frac{3}{2}x^{2}}\\cdot \\left(-\\frac{8}{5}x^{3}+x^{2}+3x\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{1}{4}x^{3}-\\frac{1}{3}x^{2}+\\frac{4}{3}x\\right)}\\cdot \\left(-\\frac{3}{4}x^{2}-\\frac{2}{3}x+\\frac{4}{3}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{4}{9}x^{3}+\\frac{3}{4}x-\\frac{1}{3}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{1}{8}x^{3}+\\frac{1}{7}x^{2}-\\frac{1}{9}x\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{4}{9}x^{3}+\\frac{3}{4}x-\\frac{1}{3}}\\cdot \\left(-\\frac{4}{3}x^{2}+\\frac{3}{4}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{1}{8}x^{3}+\\frac{1}{7}x^{2}-\\frac{1}{9}x\\right)}\\cdot \\left(\\frac{3}{8}x^{2}+\\frac{2}{7}x-\\frac{1}{9}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{1}{7}x^{3}+\\frac{1}{2}x-\\frac{1}{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{1}{3}x^{2}-\\frac{3}{2}x+\\frac{2}{3}\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{1}{7}x^{3}+\\frac{1}{2}x-\\frac{1}{2}}\\cdot \\left(\\frac{3}{7}x^{2}+\\frac{1}{2}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{1}{3}x^{2}-\\frac{3}{2}x+\\frac{2}{3}\\right)}\\cdot \\left(-\\frac{2}{3}x-\\frac{3}{2}\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(\\frac{2}{5}x^{4}+\\frac{3}{8}x^{3}+\\frac{1}{3}x\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(\\frac{1}{9}x^{4}-\\frac{2}{5}x^{3}-\\frac{3}{4}x^{2}\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{\\frac{2}{5}x^{4}+\\frac{3}{8}x^{3}+\\frac{1}{3}x}\\cdot \\left(\\frac{8}{5}x^{3}+\\frac{9}{8}x^{2}+\\frac{1}{3}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(\\frac{1}{9}x^{4}-\\frac{2}{5}x^{3}-\\frac{3}{4}x^{2}\\right)}\\cdot \\left(\\frac{4}{9}x^{3}-\\frac{6}{5}x^{2}-\\frac{3}{2}x\\right)\\quad$ "], ["a) $f(x)=\\ln\\left(-\\frac{3}{7}x^{4}-\\frac{4}{3}x^{3}+\\frac{1}{2}\\right)\\quad$ b) $f(x)=\\mathrm{e}^{\\left(-\\frac{2}{3}x^{4}+\\frac{3}{4}x+\\frac{1}{3}\\right)}\\quad$ ", "a) $f'(x)=\\frac{1}{-\\frac{3}{7}x^{4}-\\frac{4}{3}x^{3}+\\frac{1}{2}}\\cdot \\left(-\\frac{12}{7}x^{3}-4x^{2}\\right)\\quad$ b) $f'(x)=\\mathrm{e}^{\\left(-\\frac{2}{3}x^{4}+\\frac{3}{4}x+\\frac{1}{3}\\right)}\\cdot \\left(-\\frac{8}{3}x^{3}+\\frac{3}{4}\\right)\\quad$ "]], "
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