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Es werden die grundlegenden Ableitungsregeln geübt.
\nDie Reihenfolge der Summanden ist auch randomisiert.
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "Geben Sie die Ableitung f'(x) von f(x) an.
", "advice": "In dieser Aufgabe ${\\simplify[std,!collectLikeFractions]{ f(x) = ({a[2]}x ^ {n[2]}) + ({a[1]}x^{n[1]}) + ({a[0]}x^{n[0]}) } }$ sind die einzelnen Ableitungen:
\n\\[ g(x) = \\simplify[std]{{a[2]}x ^ {n[2]}} \\Rightarrow g'(x) = \\var[fractionNumbers]{a[2]} \\cdot \\var{n[2]}x ^ \\var{n[2]-1} = \\simplify[std]{{{a[2]}*{n[2]}}x ^ {n[2]-1}} \\]
\n\\[ g(x) = \\simplify[std]{{a[1]}x ^ {n[1]}} \\Rightarrow g'(x) = \\var[fractionNumbers]{a[1]} \\cdot \\var{n[1]}x ^ \\var{n[1]-1} = \\simplify[std]{{{a[1]}*{n[1]}}x ^ {n[1]-1}} \\]
\n\\[ g(x) = \\simplify[std]{{a[0]}x ^ {n[0]}} \\Rightarrow g'(x) = \\var[fractionNumbers]{a[0]} \\cdot \\var{n[0]}x ^ \\var{n[0]-1} = \\simplify[std]{{{a[0]}*{n[0]}}x ^ {n[0]-1}} \\]
\nAlso ist die Ableitung von $f$ : $\\simplify[std,!collectLikeFractions]{ f'(x) = {{a[2]}*{n[2]}}x ^ {n[2]-1} + {{a[1]}*{n[1]}}x^{n[1]-1} +{{a[0]}*{n[0]}}x^{n[0]-1} }$
", "rulesets": {"std": ["all", "!collectLikeFractions", "!collectNumbers", "fractionNumbers", "timesDot"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "shuffle(0..4)", "description": "Grad des Polynoms
", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "[random([-1,1])*random(1..6)/random(1..6),random([-1,1])*random(1..6)/random(1..6),random([-1,1])*random(1..6)/random(1..6)]", "description": "Das Polynom besteht aus drei Summanden. Dies sind die Koeffizienten.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "${\\simplify[std,!collectLikeFractions]{ f(x) = {a[2]}x ^ {n[2]} + {a[1]}x^{n[1]} + {a[0]}x^{n[0]} } }$
\n\n$\\displaystyle f'(x)=\\;$[[0]]
\n\nBitte beachten:
\nDie Ableitung einer Potenzfunktionen wird gebildet mit
\n\\[ f(x) = x^n \\Rightarrow f'(x) = n \\cdot x^{n-1} \\]
\nMit der Faktorregel
\n\\[ f(x) = a \\cdot g(x) \\Rightarrow f'(x) = a \\cdot g'(x) \\]
\nund der Summenregel
\n\\[ f(x) = g(x) + h(x) \\Rightarrow f'(x) = g'(x) + h'(x)\\]
\nkönnen Linearkombinationen von Funktionen abgeleitet werden, also auch Polynomfunktionen.
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