// Numbas version: exam_results_page_options {"question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Product rule, three questions, advice included", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "preventleave": false, "showfrontpage": false}, "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/", "name": "Lovkush Agarwal"}], "variables": {"a5": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0,1])", "name": "a5"}, "d4": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-2..-1) + b4", "name": "d4"}, "a1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0,1])", "name": "a1"}, "a3": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0,1])", "name": "a3"}, "a2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0,1])", "name": "a2"}, "a6": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0,1])", "name": "a6"}, "c4": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0,1])", "name": "c4"}, "b4": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5 except [-1,0,1])", "name": "b4"}, "a4": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0,1])", "name": "a4"}, "b6": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "b6"}, "b2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5 except [-1,0,1])", "name": "b2"}}, "ungrouped_variables": ["a1", "a2", "b2", "a3", "a4", "b4", "c4", "d4", "a5", "a6", "b6"], "variable_groups": [], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International", "description": "

Differentiating three functions which require using the product rule. Non-calculator. Advice included.

(Note, these are not 'model' answers, but explanations of how I obtain an answer. You do not need this much explanation in your answers.)

\n

\n

\n

(i) Differentiating $\\var{a2}x^{\\var{b2}}$ gives $\\simplify{{a2}*{b2}*x^({b2}-1)}$, so we get $\\simplify{{a2}*{b2}*x^({b2}-1)*cos(x)}$ as one of the terms.

\n

Differentiating $\\cos(x)$ gives $-\\sin(x)$ so the other term is $\\simplify{- {a2}*x^({b2})*sin(x)}$.

\n

Adding these together gives the answer: $\\simplify{{a2}*{b2}*x^({b2}-1)*cos(x) - {a2}*x^({b2})*sin(x)}$.

\n

\n

\n

(ii) Differentiating $\\simplify{{a4}*x^{b4} + {c4}*x^{d4}}$ gives $\\simplify{{a4}*{b4}*x^({b4}-1) + {c4}*{d4}*x^({d4}-1)}$, so we get $\\simplify{({a4}*{b4}*x^({b4}-1) + {c4}*{d4}*x^({d4}-1))ln(x)}$ as one of the terms.

\n

Differentiating $\\simplify{ln(x)}$ gives $\\simplify{1/x}$ so the other term is $\\simplify{({a4}*x^{b4} + {c4}*x^{d4})/x}$, which simplifies to $\\simplify{{a4}*x^({b4}-1) + {c4}*x^({d4}-1)}$

\n

Adding these together gives the answer: $\\simplify{({a4}*{b4}*x^({b4}-1) + {c4}*{d4}*x^({d4}-1))ln(x) + {a4}*x^({b4}-1) + {c4}*x^({d4}-1)}$.

\n

\n

\n

\n

(iii) Differentiating $\\simplify{ln(x)}$ gives $\\simplify{1/x}$, so we get $\\simplify{{a5}*sin(x)/x}$ as one of the terms.

\n

Differentiating $\\simplify{{a5}*sin(x)}$ gives $\\simplify{{a5}*cos(x)}$ so the other term is $\\simplify{{a5}*cos(x)*ln(x)}$.

\n

Adding these together gives the answer: $\\simplify{{a5}*sin(x)/x + {a5}*cos(x)*ln(x)}$.

", "statement": "

Use the product rule to differentiate.

", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showFeedbackIcon": true, "gaps": [{"showFeedbackIcon": true, "checkVariableNames": false, "answerSimplification": "all", "failureRate": 1, "answer": "{a2}*{b2}*x^({b2}-1)*cos(x) - {a2}*x^({b2})*sin(x)", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "vsetRange": [0, 1], "marks": "2", "showCorrectAnswer": true, "showPreview": true, "checkingAccuracy": 0.001, "type": "jme", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "scripts": {}}, {"showFeedbackIcon": true, "checkVariableNames": false, "answerSimplification": "all", "failureRate": 1, "answer": "({a4}*{b4}*x^({b4}-1) + {c4}*{d4}*x^({d4}-1))ln(x) + {a4}*x^({b4}-1) + {c4}*x^({d4}-1)", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "vsetRange": [0, 1], "marks": "2", "showCorrectAnswer": true, "showPreview": true, "checkingAccuracy": 0.001, "type": "jme", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "scripts": {}}, {"showFeedbackIcon": true, "checkVariableNames": false, "answerSimplification": "all", "failureRate": 1, "answer": "{a5}*sin(x)/x + {a5}*cos(x)*ln(x)", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "vsetRange": [0, 1], "marks": "2", "showCorrectAnswer": true, "showPreview": true, "checkingAccuracy": 0.001, "type": "jme", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "scripts": {}}], "variableReplacementStrategy": "originalfirst", "unitTests": [], "variableReplacements": [], "type": "gapfill", "marks": 0, "prompt": "

Differentiate the following with respect to $x$.

\n

(i) $\\var{a2}x^{\\var{b2}} \\cos(x)$ [[0]]

\n

(ii) $(\\simplify{{a4}*x^{b4} + {c4}*x^{d4}})\\ln(x)$ [[1]]

\n

(iii) $\\var{a5}\\sin(x) \\ln(x)$ [[2]]

", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "scripts": {}}]}], "name": "Group"}], "duration": 0, "custom_part_types": [], "percentPass": 0, "resources": [], "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Questions on differentiating with the chain rule and product rule.

"}, "type": "exam", "extensions": [], "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/", "name": "Blathnaid Sheridan"}], "showQuestionGroupNames": false, "name": "Blathnaid's copy of CLE 6a", "feedback": {"intro": "", "showactualmark": true, "advicethreshold": 0, "showanswerstate": true, "showtotalmark": true, "allowrevealanswer": true, "feedbackmessages": []}, "navigation": {"showresultspage": "oncompletion", "allowregen": false, "onleave": {"action": "none", "message": ""}, "showfrontpage": true, "reverse": true, "preventleave": true, "browse": true}}