// Numbas version: finer_feedback_settings
{"name": "Logaritmer", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": "80", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Forenkling av uttrykk", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Logaritmer - forenkle uttrykk 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Torris Bakke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/942/"}], "tags": [], "metadata": {"description": "<p>Expand $\\ln(a^{x}b^{y})$ into $x\\ln(a)+y\\ln(b)$, x and y randomised.</p>", "licence": "None specified"}, "statement": "<p>Forenkle uttrykkene mest mulig.</p>", "advice": "<p>Her bruker vi f&oslash;rste og tredje logaritmesetning:</p>\n<ul>\n<li>F&oslash;rste: $\\ln(ab)=\\ln(a)+\\ln(b)$<br/><br/></li>\n<li>Tredje: $\\ln(a^{b})=b\\ln(a)$</li>\n</ul>\n<p>Da f&aring;r vi at $\\ln(a^{\\var{c[0]}}b^{\\var{c[1]}})=\\var{c[0]}\\ln(a)+\\var{c[1]}\\ln(b)$.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..7),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c"], "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>$\\ln(a^{\\var{c[0]}}b^{\\var{c[1]}})=$[[0]]$\\ln(a)+$[[1]]$\\ln(b)$</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[0]", "maxValue": "c[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[1]", "maxValue": "c[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Logaritmer - forenkle uttrykk 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Torris Bakke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/942/"}], "tags": [], "metadata": {"description": "<p>Expand $\\ln(\\dfrac{a^{x}}{b^{y}})$ into $x\\ln(a)-y\\ln(b)$, x and y randomised.</p>", "licence": "None specified"}, "statement": "<p>Forenkle uttrykkene mest mulig.</p>", "advice": "<p>Her bruker vi andre og tredje logaritmesetning:</p>\n<ul>\n<li>Andre: $\\ln(\\frac{a}{b})=\\ln(a)-\\ln(b)$<br/><br/></li>\n<li>Tredje: $\\ln(a^{b})=b\\ln(a)$</li>\n</ul>\n<p>Da f&aring;r vi at $\\ln(\\frac{a^{\\var{c[0]}}}{b^{\\var{c[1]}}})=\\var{c[0]}\\ln(a)-\\var{c[1]}\\ln(b)$.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..7),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c"], "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>$\\ln(\\dfrac{a^{\\var{c[0]}}}{b^{\\var{c[1]}}})=$[[0]]$\\ln(a)+$[[1]]$\\ln(b)$</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[0]", "maxValue": "c[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-c[1]", "maxValue": "-c[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Logaritmer - forenkle uttrykk 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Torris Bakke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/942/"}], "tags": [], "metadata": {"description": "<p>Simlify (expand) logarithmic expressions&nbsp;</p>", "licence": "None specified"}, "statement": "<p>Forenkle uttrykket.</p>", "advice": "<p>Her bruker vi logaritmesetningene:</p>\n<ol>\n<li>$\\ln(ab)=\\ln(a)+\\ln(b)$</li>\n<li>$\\ln\\left(\\dfrac{a}{b}\\right)=\\ln(a)-\\ln(b)$</li>\n<li>$\\ln\\left(a^{b}\\right)=b\\ln(a)$</li>\n</ol>\n<p>I tillegg bruker vi sammenhengen $\\ln(e)=1$.</p>\n<p>Vi skal forenkle uttrykket</p>\n<p>\\[\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)-\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)+\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)\\]</p>\n<p>Her kan det v&aelig;re en fordel &aring; forenkle hvert ledd for seg f&oslash;rst:</p>\n<ul>\n<li>$\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)=\\ln(\\var{b1})+\\ln(a^{\\var{c[0]}})=\\ln(2^{\\var{ex[0]}})+\\var{c[0]}\\ln(a)=\\var{ex[0]}\\ln(2)+\\var{c[0]}\\ln(a)$</li>\n<li>$\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)=\\ln(b^{\\var{c[1]}})-\\ln(a^{\\var{c[2]}})=\\var{c[1]}\\ln(b)-\\var{c[2]}\\ln(a)$</li>\n<li>$\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)=\\ln(e^{\\var{c[3]}})-\\ln(\\var{b2}b^{\\var{c[4]}})=\\var{c[3]}\\ln(e)-\\left(\\ln(\\var{b2})+\\ln(b^{\\var{c[4]}})\\right)=\\var{c[3]}-\\ln(2^{\\var{ex[1]}})-\\ln(b^{\\var{c[4]}})=\\var{c[3]}-\\var{ex[1]}\\ln(2)-\\var{c[4]}\\ln(b)$</li>\n</ul>\n<p>S&aring; setter vi sammen:</p>\n<p>$\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)-\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)+\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)\\\\<br/>=\\var{ex[0]}\\ln(2)+\\var{c[0]}\\ln(a)-\\left(\\var{c[1]}\\ln(b)-\\var{c[2]}\\ln(a)\\right)+\\var{c[3]}-\\var{ex[1]}\\ln(2)-\\var{c[4]}\\ln(b)\\\\<br/>=\\var{c[0]}\\ln(a)+\\var{c[2]}\\ln(a)-\\var{c[1]}\\ln(b)-\\var{c[4]}\\ln(b)+\\var{ex[0]}\\ln(2)-\\var{ex[1]}\\ln(2)+\\var{c[3]}\\\\<br/>=\\underline{\\underline{\\var{ka}\\ln(a)-\\var{kb}\\ln(b)+\\var{k2}\\ln(2)+\\var{c[3]}}}$</p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..7),6)", "description": "", "templateType": "anything", "can_override": false}, "ex": {"name": "ex", "group": "Ungrouped variables", "definition": "repeat(random(1..5),2)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "2^(ex[0])", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "2^(ex[1])", "description": "", "templateType": "anything", "can_override": false}, "ka": {"name": "ka", "group": "Ungrouped variables", "definition": "c[0]+c[2]", "description": "", "templateType": "anything", "can_override": false}, "kb": {"name": "kb", "group": "Ungrouped variables", "definition": "c[1]+c[4]", "description": "", "templateType": "anything", "can_override": false}, "k2": {"name": "k2", "group": "Ungrouped variables", "definition": "ex[0]-ex[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "ex", "b1", "b2", "ka", "kb", "k2"], "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>$\\ln\\left(\\var{b1}a^{\\var{c[0]}}\\right)-\\ln\\left(\\dfrac{b^{\\var{c[1]}}}{a^{\\var{c[2]}}}\\right)+\\ln\\left(\\dfrac{e^{\\var{c[3]}}}{\\var{b2}b^{\\var{c[4]}}}\\right)$=[[0]]$\\ln(a)-$[[1]]$\\ln(b)+$[[2]]$\\ln(2)+$[[3]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[0]+c[2]", "maxValue": "c[0]+c[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[1]+c[4]", "maxValue": "c[1]+c[4]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex[0]-ex[1]", "maxValue": "ex[0]-ex[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c[3]", "maxValue": "c[3]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Likninger", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Logaritme- og eksponentiallikninger", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Torris Bakke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/942/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>L&oslash;s likningene. Avrund svaret til 2 desimaler.</p>", "advice": "<p><strong>a)</strong></p>\n<p>L&oslash;sningsformelen for en logaritmelikning p&aring; grunnform ser slik ut:</p>\n<p>$\\begin{align}\\ln(x)&amp;=a\\\\ &amp;\\Downarrow \\\\x&amp;=e^{a}\\end{align}$</p>\n<p>Vi snur p&aring; likningen og regner ut:</p>\n<p>$\\begin{align}<br/>\\simplify{{a1} ln(x)+{b1}}&amp;=\\var{c1}\\\\<br/>\\simplify{{a1}ln(x)} &amp;= \\simplify[!collectNumbers]{{c1}-{b1}}\\\\<br/>\\simplify{{a1}ln(x)}&amp;= \\simplify{{c1}-{b1}}\\\\<br/>\\ln(x) &amp;= \\simplify{{{c1}-{b1}}/{{a1}}}\\\\<br/>x &amp;= e^{\\simplify{{{c1}-{b1}}/{{a1}}}}\\\\<br/>\\end{align}$</p>\n<p>$x=${precround({exp(({c1}-{b1})/{a1})},2)}</p>\n<p><strong>b)</strong></p>\n<p>Likningen $\\simplify{{a2}(ln(x))^{2}+{b2}ln(x)+{c2}=0}$ er en andregradslikning med $\\ln(x)$ som ukjent. Vi bruker da f&oslash;rst <em>abc</em>-formelen for &aring; finne $\\ln(x)$:</p>\n<p>$a=\\var{a2},\\;b=\\var{b2},\\;c=\\var{c2}$</p>\n<p>$\\ln(x)=\\dfrac{\\simplify{-{b2}}\\pm\\sqrt{\\simplify[!collectNumbers,timesDot]{{b2}^{2}-4*{a2}*{c2}}}}{2\\cdot\\var{a2}}=\\dfrac{\\simplify{-{b2}}\\pm\\sqrt{\\simplify{{b2}^2-4*{a2}*{c2}}}}{\\simplify{2*{a2}}}$.</p>\n<p>Vi to l&oslash;sninger for $\\ln(x)$: $\\ln(x)= \\var{x[0]}\\text{ og } \\ln(x)=\\var{x[1]}$.</p>\n<p>Vi l&oslash;ser for $x$:</p>\n<p>$\\ln(x)= \\var{x[0]}\\Rightarrow x=e^{\\var{x[0]}}=${precround(e^(x[0]),6)}$\\approx\\underline{\\underline{\\var{sol[0]}}}$</p>\n<p>$\\ln(x)= \\var{x[1]}\\Rightarrow x=e^{\\var{x[1]}}=${precround(e^(x[1]),6)}$\\approx\\underline{\\underline{\\var{sol[1]}}}$</p>\n<p></p>\n<p><strong>c)</strong></p>\n<p>L&oslash;sningsformelen for eksponentiallikning p&aring; grunnform ser slik ut:</p>\n<p>$\\begin{align}e^{x}&amp;=a\\\\ &amp;\\Downarrow \\\\x&amp;=\\ln(a)\\end{align}$</p>\n<p>Vi snur p&aring; likningen og regner ut:</p>\n<p>$\\begin{align}<br/>e^{\\simplify{{c1}x}}-\\var{e1}&amp;=0\\\\<br/>e^{\\simplify{{c1}x}}&amp;=\\var{e1}\\\\<br/>\\var{c1}x &amp;= \\ln(\\var{e1})\\\\<br/>x &amp;=\\dfrac{\\ln(\\var{e1})}{\\var{c1}}\\\\<br/>\\end{align}$</p>\n<p>$x=${precround(ln(e1)/c1,2)}</p>\n<p></p>\n<p><strong>d)</strong></p>\n<p>Likningen $\\simplify{{a2}e^(2*x)+{c2}e^(x)+{b2}=0}$ er en andregradslikning med $e^{x}$ som ukjent. Vi bruker da f&oslash;rst <em>abc</em>-formelen for &aring; finne $e^{x}$:</p>\n<p>$a=\\var{a2},\\;b=\\var{c2},\\;c=\\var{b2}$</p>\n<p>$e^{x}=\\dfrac{\\simplify{-{c2}}\\pm\\sqrt{\\simplify[!collectNumbers,timesDot]{{c2}^{2}-4*{a2}*{b2}}}}{2\\cdot\\var{a2}}=\\dfrac{\\simplify{-{c2}}\\pm\\sqrt{\\simplify{{c2}^2-4*{a2}*{b2}}}}{\\simplify{2*{a2}}}$.</p>\n<p>Vi to l&oslash;sninger for $e^{x}$: $e^{x}= \\var{x2[0]}\\text{ og } e^{x}=\\var{x2[1]}$.</p>\n<p>$e^{x}= \\var{x2[0]}\\Rightarrow x=\\ln(\\var{x2[0]})\\approx\\underline{\\underline{\\var{sol2[0]}}}$</p>\n<p>$e^{x}= \\var{x2[1]}\\Rightarrow x=\\ln(\\var{x2[1]})\\approx\\underline{\\underline{\\var{sol2[1]}}}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-4..4 except 0) ", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": 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{"description": "", "licence": "None specified"}, "statement": "<p>Deriver uttrykkene.</p>", "advice": "<p><strong>a)</strong></p>\n<p>Her er det lurt &aring; forenkle uttrykket&nbsp;<strong>f&oslash;r</strong> vi deriverer. Vi bruker logaritmesetningene:</p>\n<p>$y=\\simplify{ln(x^{a})+{b}ln({c}x)}=\\simplify{{a}ln(x)}+\\simplify{{b}(ln({c})+ln(x))}=\\simplify{({a}+{b})ln(x)+{b}ln({c})}$.</p>\n<p>Vi kan da derivere leddvis, og f&aring;r</p>\n<p>\\[ y'=(\\simplify{({a}+{b})ln(x)})'+(\\var{b}ln(\\var{c}))'=\\simplify{({a}+{b})/x}+0=\\underline{\\underline{\\simplify{({a}+{b})/x}}}\\]</p>\n<p></p>\n<p><strong>b)</strong></p>\n<p>Her b&oslash;r vi ogs&aring; forenkle f&oslash;rst:</p>\n<p>$y=\\simplify{ln(x^{a}/(sqrt(x)))}=\\ln(x^{\\var{a}}-\\simplify{ln(sqrt(x))}=\\var{a}\\ln(x)-\\ln(x^{\\frac{1}{2}})=\\var{a}\\ln(x)-\\frac{1}{2}\\ln(x)=\\simplify{((2*{a}-1)/2)}\\ln(x)$.</p>\n<p>Da f&aring;r vi:</p>\n<p>\\[y'=(\\simplify{((2*{a}-1)/2)}\\ln(x))'=\\underline{\\underline{\\simplify{((2*{a}-1)/(2x))}}}\\]</p>\n<p></p>\n<p><strong>c)</strong></p>\n<p>Det finnes ingen regler for &aring; forenkle logaritmen n&aring;r argumentet er en <strong>sum.</strong></p>\n<p>Her l&oslash;nner det seg &aring; huske \"snarveien\" $\\left(\\ln(u)\\right)=\\dfrac{u'}{u}$. Vi trenger da bare &aring; velge $u$ og regne ut $u'$:</p>\n<p>$u=\\simplify[unitFactor,!collectNumbers]{x^{d}+{e}x}, \\qquad u'=\\simplify{{d}x^{{d}-1}+{e}}$</p>\n<p>Vi f&aring;r da at $y'=\\dfrac{u'}{u}=\\underline{\\underline{\\dfrac{\\simplify{{d}x^{{d}-1}+{e}}}{\\simplify{x^{d}+{e}x}}}}$.</p>\n<p></p>\n<p>Ellers m&aring; vi bruke kjerneregelen fullt ut:</p>\n<p>$u=\\simplify[unitFactor,!collectNumbers]{x^{d}+{e}x}, \\qquad u'=\\simplify{{d}x^{{d}-1}+{e}}$</p>\n<p>$y(u)=\\ln(u), \\qquad y'(u)=\\dfrac{1}{u}$</p>\n<p>$y'=y'(u)\\cdot u' = \\dfrac{1}{u}\\cdot\\simplify{{d}x^{{d}-1}+{e}}=\\dfrac{\\simplify{{d}x^{{d}-1}+{e}}}{\\simplify{x^{d}+{e}x}}$</p>\n<p></p>\n<p><strong>d)</strong></p>\n<p>Her m&aring; vi bruke kjerneregelen:</p>\n<p>$u=\\simplify{x^{d}+x^{f}}, \\qquad u'=\\simplify{{d}x^{{d}-1}+{f}x^{{f}-1}}$</p>\n<p>$y(u)=e^{u}, \\qquad y'(u)=e^{u}$</p>\n<p>$y'=y'(u)\\cdot u'=e^{u}\\cdot 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