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{"name": "Ableitung von Exponentialfunktionen mit linearen Exponenten", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ableitung von Exponentialfunktionen mit linearen Exponenten", "tags": [], "metadata": {"description": "<p>Es werden grundlegenden Ableitungsregeln ge&uuml;bt.</p>\n<p>Die Reihenfolge der Summanden ist auch randomisiert.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Geben Sie die Ableitung f'(x) von f(x) an.</p>", "advice": "<p></p>\n<p>In dieser Aufgabe&nbsp;<br/>$ f(x) =&nbsp;\\simplify[std,!otherNumbers,!noLeadingMinus]{<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {z[2]}*{c}e^({b[1]}*x+{b[0]}) &nbsp;+ {a[1]}x^{n[1]} + <br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {z[1]}*{c}e^({b[1]}*x+{b[0]})&nbsp; + {a[0]}x^{n[0]} +&nbsp;<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {z[0]}*{c}e^({b[1]}*x+{b[0]}) &nbsp; }$<br/> sind die einzelnen Ableitungen:</p>\n<p>\\[ 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} } \\]</p>\n<p>\\[ 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} } \\]</p>\n<p>\\[ g(x) = \\simplify[std,!noLeadingMinus]{{c}e^({b[1]}*x+{b[0]}) } \\Rightarrow <br/>&nbsp;&nbsp; g'(x) = (\\var[fractionNumbers]{c}) \\cdot&nbsp; (\\var{b[1]}) \\cdot \\simplify[std,!noLeadingMinus]{e^({b[1]}*x+{b[0]}) } = <br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; \\simplify[std,!noLeadingMinus]{{c}*{b[1]}*e^({b[1]}*x+{b[0]}) } \\]</p>\n<p>Also ist die Ableitung von $f$ : <br/>$ f(x) =&nbsp;\\simplify[std,!otherNumbers,!noLeadingMinus]{<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {c}*{b[1]}e^({b[1]}*x+{b[0]}) &nbsp;+&nbsp;{{a[1]}*{n[1]}}x ^ {n[1]-1} &nbsp; +&nbsp;{{a[0]}*{n[0]}}x ^ {n[0]-1} &nbsp; }$</p>", "rulesets": {"std": ["all", "!collectLikeFractions", "fractionNumbers", "timesDot"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "shuffle(0..2)", "description": "<p>Grad des Polynoms</p>", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "[random([-1,1])*random(1..6)/random(1..6),random([-1,1])*random(1..6)/random(1..6)]", "description": "<p>Die Koeffizienten f&uuml;r den polynomialen Teil.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random([-1,1])*random(1..6)/random(1..6)", "description": "<p>Koeffizient der Exponentialfunktion.</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "[random(-1 .. 1), random([-4,-3,-2,-1,1,2,3,4])]", "description": "<p>Koeffizienten des linearen Exponenten</p>", "templateType": "anything", "can_override": false}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "shuffle([1,0,0])", "description": "<p>Position der e-Funktion in der Summe</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "a", "c", "b", "z"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$ f(x) =&nbsp;\\simplify[std,!otherNumbers,!noLeadingMinus]{<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {z[2]}*{c}e^({b[1]}*x+{b[0]}) &nbsp;+ {a[1]}x^{n[1]} + <br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {z[1]}*{c}e^({b[1]}*x+{b[0]})&nbsp; + {a[0]}x^{n[0]} +&nbsp;<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; {z[0]}*{c}e^({b[1]}*x+{b[0]}) &nbsp; }$</p>\n<p></p>\n<p>$\\displaystyle f'(x)=\\;$[[0]]</p>\n<p></p>\n<p><strong>Bitte beachten:</strong></p>\n<ul>\n<li>Br&uuml;che mit <strong>Slash /</strong> eingeben: 3/2 f&uuml;r $\\frac{3}{2}$&nbsp;</li>\n<li>Exponenten mit <strong>^</strong> eingeben: x^2 f&uuml;r $x^2$<br/>Nach der <strong>^-Taste</strong> m&uuml;ssen Sie eventuell noch die <strong>Leertaste</strong> dr&uuml;cken.</li>\n<li>Mehrteilige Exponenten mit <strong>runden Klammern</strong> umschlie&szlig;en: e^(2x) f&uuml;r $e^{2x}$</li>\n<li>Dezimalzahlen mit <strong>Punkt</strong> eingeben: $3.14$ statt $3,14$ .</li>\n</ul>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": true, "customName": "Ableitungsregeln", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Die Ableitung einer <strong>Potenzfunktionen</strong> wird gebildet mit</p>\n<p>\\[ f(x) = x^n \\Rightarrow f'(x) = n \\cdot x^{n-1}&nbsp; \\]</p>\n<p>Die Ableitung einer <strong>Exponentialfunktion</strong> erfolgt mit die Kettenregel</p>\n<p>\\[ f(x) = e^{kx+b} \\Rightarrow f'(x) = k \\cdot e^{kx+b} \\]</p>\n<p>Mit der <strong>Faktorregel</strong></p>\n<p>\\[ f(x) = a \\cdot g(x) \\Rightarrow f'(x) = a \\cdot g'(x) \\]</p>\n<p>und der <strong>Summenregel</strong></p>\n<p>\\[ f(x) =&nbsp; g(x) + h(x) \\Rightarrow f'(x) = g'(x) + h'(x)\\]</p>\n<p>k&ouml;nnen Linearkombinationen von Funktionen abgeleitet werden.</p>\n<p></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b[1]}*{c}e^({b[1]}*x+{b[0]})  + {a[1]}*{n[1]}x^({n[1]}-1) + {a[0]}*{n[0]}x^({n[0]}-1)", "answerSimplification": "std,!noLeadingMinus", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Helge M\u00fcnnich", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1885/"}]}]}], "contributors": [{"name": "Helge M\u00fcnnich", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1885/"}]}