// Numbas version: finer_feedback_settings
{"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Product rule", "tags": ["Calculus", "Steps", "calculus", "derivative of a product", "differentiating a product", "differentiating exponential functions", "differentiating trigonometric functions", "differentiation", "product rule", "steps"], "advice": "\n  \n  \n  <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  \n  \n  \n  <p>For this example:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{u = sin({a} + {b} * x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {b} * cos({a} + {b} * x)}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = e ^ ({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the product rule above we get:</p>\n  \n  \n  \n  <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  \n  &#38;=&#38;\\simplify[std]{({b}cos({a}+{b}x)+{n}sin({a}+{b}x))e^({n}x)}\n  \n  \\end{eqnarray*}\\]</p>\n  \n  \n    ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "<p>$\\simplify[std]{f(x) = sin({b}x + {a}) * e ^ ({n} * x)}$</p>\n      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      <p>Click on <em>Show steps</em> for more information. You will not lose any marks by doing so.</p>", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{b} * cos({a} + {b} * x) * e ^ ({n} * x) + {n} * sin({a} + {b} * x) * e ^ ({n} * x)", "type": "jme"}], "steps": [{"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>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "b": {"definition": "s1*random(2..9)", "name": "b"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "random(2..8)", "name": "m"}, "n": {"definition": "s2*random(2..5)", "name": "n"}}, "metadata": {"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>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}