// Numbas version: exam_results_page_options
{"name": "tg_TR_derivasjonsregler_1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "n", "a1", "a2", "b1", "b2", "c1", "n2", "t"], "name": "tg_TR_derivasjonsregler_1", "tags": ["chain rule", "derivative", "Differentiation", "differentiation"], "preamble": {"css": "", "js": ""}, "advice": "<p>1) Den deriverte av $\\;\\simplify {f(x) = {a} x + {b}}\\;$ finner vi ved &aring; kombinere reglene (1), (2) og (3), men vi kan ogs&aring; se at grafen til funksjonen er en rett linje med konstant stigningstall {a}, og det blir verdien av den deriverte.<br/>Etter regel (1) og (3) blir den deriverte av en konstant $b$ lik null fordi vi kan skrive $b = b \\cdot 1 = b \\cdot x^0$ , og deriverer vi $f(x)=x^0$ etter regel (1) f&aring;r vi $f'(x) = 0 \\cdot x^{-1} = 0$ - det skulle bare mangle, en konstant har en linje parallell med x-aksen som graf, med stigningstall null. Den deriverte av $\\; {f(x) = \\var{a} x = \\var{a} x^1}\\;$ blir etter produktregel$\\;{f'(x) =\\var{a} x = \\var{a} \\cdot 1 \\cdot x^0 = \\var{a}}\\;$<br/>Kort sagt, den deriverte av $\\;f(x)=a \\cdot x + b\\;$ er $\\;f'(x)=a\\;$.</p>\n<p>2) Funksjonsuttrykket $\\;\\simplify {f(x) = {a1} x^2 + {b1}x + {c1}}\\;$ er en sum av tre ledd som deriveres hver for seg, $\\;f'(x)= \\var{a1} \\cdot (x^2)' + \\var{b1} \\cdot (x)' + \\var {c1} \\cdot (x^0)' = 2 \\cdot \\var{a1} x^1 + 1 \\cdot \\var {b1} \\cdot x^0 + 0 \\cdot \\var{c1} \\cdot x^{-1} = \\var {a1} x + \\var{b1}$</p>\n<p>3) Den deriverte av $\\;{f(x) = \\frac {x^\\var{a2}} {\\var{b2}} - \\frac {\\var{b2}} {x^{\\var{a2}}} = \\frac 1 {\\var{b2}} \\cdot x^\\var{a2} - \\var {b2} \\cdot x^{- \\var{a2}} }\\;$ deriveres som $\\;f'(x) = \\frac 1 {\\var{b2}} \\cdot \\var {a2} \\cdot x^{\\var{a2} - 1} - \\var {b2} \\cdot (- \\var{a2}) \\cdot x^{-\\var {a2}-1}$</p>\n<p>5) Her kan produktregelen brukes, men hvorfor ikke utvide uttrykket f&oslash;rst?<br/>$f(x) = (\\var{a}-x^{\\var{b}})x^\\frac {1} {\\var{n}} = \\var{a}x^\\frac {1} {\\var{n}} - x^{\\simplify{{b}+1/{n}}}$</p>\n<p>6) Br&oslash;kfunksjonen $\\;f(x) = \\frac {\\simplify{x^{n}-{a1}}}{\\simplify{{n} - x^{n2}}} = \\frac u v\\;$ har $\\;u' = \\var {n} \\cdot x^{\\var{n}-1}\\;$ og $\\;v' = - \\var {n2} \\cdot x^{\\var{n2}-1}\\;$ som settes inn i br&oslash;kregelen, $\\;(\\frac u v)' = \\frac {u' \\cdot v - u \\cdot v'} {v^2}$</p>\n<p>7) Funksjonen $\\;f(x) = \\var{a2}(x^{\\var{b2}}-\\var{n})^\\var{n2} =\\var{a2}(u)^\\var{n2} \\;$ har $\\; u = x^{\\var{b2}}-\\var{n}\\;$ som kjernefunksjon, med derivert $\\;u' = \\var {b2} \\cdot x^{\\var{b2}-1}\\;$<br/>Den deriverte blir $\\;f'(x) = \\var{a2} \\cdot&nbsp; \\var {n2} \\cdot (x^{\\var{b2}}-\\var{n})^{\\var{n2}-1} \\cdot \\var {b2}\\cdot x^{\\var{b2}-1}\\;$ .</p>", "rulesets": {}, "parts": [{"prompt": "<p>1) Den deriverte av $\\;\\simplify {f(x) = {a} x + {b}}\\;$ er $\\;f'(x)\\;$ = [[0]]</p>\n<p>2) Den deriverte av $\\;\\simplify {f(x) = {a1} x^2 + {b1}x + {c1}}\\;$ er $\\;f'(x)\\;$ = [[1]]</p>\n<p>3) Den deriverte av $\\;{f(x) = \\frac {x^\\var{a2}} {\\var{b2}} - \\frac {\\var{b2}} {x^{\\var{a2}}}}\\;$ er $\\;f'(x)\\;$ = [[2]]</p>\n<p>4) Den deriverte av $\\;f(x) = x^\\frac {\\var{t}} {\\var{n}}\\;$ er $\\;f'(x)\\;$ = [[3]]</p>\n<p>5) Den deriverte av $\\;f(x) = (\\var{a}-x^{\\var{b}})x^\\frac {1} {\\var{n}}\\;$ er $\\;f'(x)\\;$ = [[4]]</p>\n<p>6) Den deriverte av $\\;f(x) = \\frac {\\simplify{x^{n}-{a1}}}{\\simplify{{n} - x^{n2}}}\\;$ er $\\;f'(x)\\;$ = [[5]]</p>\n<p>7) Den deriverte av $\\;f(x) = \\var{a2}(x^{\\var{b2}}-\\var{n})^\\var{n2}\\;$ er $\\;f'(x)\\;$ = [[6]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "2*{a1}x+{b1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a2}/{b2}x^({a2}-1)+{a2}*{b2}x^(-{a2}-1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{t}/{n}*x^({t}/{n}-1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "-{b}x^({b}-1)*x^(1/{n})+({a}-x^{b})/{n}*x^(1/{n}-1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({n}x^({n}-1)({n}-x^{n2})-(x^{n}-{a1})(-{n2})x^({n2}-1))/({n}-x^{n2})^2", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a2}{n2}(x^{b2}-{n})^({n2}-1){b2}x^({b2}-1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>De grunnleggende derivasjonsreglene</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: left;\"><b>Regel</b></td>\n<td style=\"text-align: left;\"><b>Funksjon, </b> $f(x)$</td>\n<td style=\"text-align: left;\"><b>Derivert, </b> $f'(x)$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">x-i-r-regelen</td>\n<td style=\"text-align: left;\">$x^r$</td>\n<td style=\"text-align: left;\">$r \\cdot x^{r-1}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">Sum av funksjoner</td>\n<td style=\"text-align: left;\">$u(x)+v(x)$</td>\n<td style=\"text-align: left;\">$u'(x)+v'(x)$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">Konstant ganger funksjon</td>\n<td style=\"text-align: left;\">$k \\cdot f(x)$</td>\n<td style=\"text-align: left;\">$k \\cdot f'(x)$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">Produkt av funksjoner</td>\n<td style=\"text-align: left;\">$u \\cdot v$</td>\n<td style=\"text-align: left;\">$u' \\cdot v + u \\cdot v'$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">Br&oslash;k av funksjoner</td>\n<td style=\"text-align: left;\">$\\displaystyle {\\frac u v}$</td>\n<td style=\"text-align: left;\">$\\displaystyle {\\frac {u' \\cdot v - u \\cdot v'} {v^2}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: left;\">Kjerneregelen</td>\n<td style=\"text-align: left;\">$\\displaystyle{g(u(x))}$</td>\n<td style=\"text-align: left;\">$\\displaystyle{g'(u(x)) \\cdot u'(x)}$</td>\n</tr>\n</tbody>\n</table>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-5..5 except [0, 1])", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-5..5 except [0, 1])", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "n": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "a1": {"definition": "random(-5..5 except [0, 1])", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "b1": {"definition": "random(-5..5 except [0, 1])", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(2..5 except a2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "c1": {"definition": "random(-5..5 except [0, 1])", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "n2": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "t": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}}, "metadata": {"description": "<p>Basic rules of derivatives</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Tore Gaupseth", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/28/"}]}]}], "contributors": [{"name": "Tore Gaupseth", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/28/"}]}