// Numbas version: exam_results_page_options
{"name": "Helge's copy of Differentiation: Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "b", "s2", "s1", "m", "n"], "parts": [{"scripts": {}, "steps": [{"scripts": {}, "prompt": "<p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>", "showCorrectAnswer": true, "marks": 0, "type": "information"}], "marks": 0, "showCorrectAnswer": true, "prompt": "\n\t\t\t<p>$\\simplify[std]{f(x) = sin({b}x + {a}) * e ^ ({n} * x)}$</p>\n\t\t\t      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n\t\t\t      <p>Click on Show steps for more information, you will not lose any marks by doing &nbsp;so.</p>\n\t\t\t", "stepsPenalty": 0, "type": "gapfill", "gaps": [{"showCorrectAnswer": true, "vsetrangepoints": 5, "expectedvariablenames": [], "checkvariablenames": false, "marks": 3, "checkingaccuracy": 0.001, "scripts": {}, "answersimplification": "std", "answer": "{b} * cos({a} + {b} * x) * e ^ ({n} * x) + {n} * sin({a} + {b} * x) * e ^ ({n} * x)", "showpreview": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "type": "jme"}]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n\t\t<p><strong>31/07/2012:</strong></p>\n\t\t<p>Checked calculation.</p>\n\t\t<p>Added tags.</p>\n\t\t<p>Allowed no penalty on looking at Show steps.</p>\n\t\t", "description": "<p>Differentiate $ \\sin(ax+b) e ^ {nx}$.</p>"}, "variable_groups": [], "name": "Helge's copy of Differentiation: Product rule", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "tags": ["calculus", "Calculus", "checked2015", "derivative of a product", "differentiating a product", "differentiating exponential functions", "differentiating trigonometric functions", "differentiation", "mas1601", "MAS1601", "product rule", "Steps", "steps"], "variables": {"a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "a", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..5)", "name": "n", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "b", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..8)", "name": "m", "description": ""}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}}, "type": "question", "advice": "\n\t  \n\t  \n\t  <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>For this example:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[std]{u = sin({a} + {b} * x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {b} * cos({a} + {b} * x)}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[std]{v = e ^ ({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>Hence on substituting into the product rule above we get:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\begin{eqnarray*}\\frac{df}{dx} &#38;=&#38; \\simplify[std]{{b} * cos({a} + {b} * x) * e ^ ({n} * x) + {n} * sin({a} + {b} * x) * e ^ ({n} * x)}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{({b}cos({a}+{b}x)+{n}sin({a}+{b}x))e^({n}x)}\n\t  \n\t  \\end{eqnarray*}\\]</p>\n\t  \n\t  \n\t", "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "showQuestionGroupNames": false, "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "functions": {}, "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/"}]}