// 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": [], "advice": "\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    <p>For this example:<br/>\\[\\simplify[std]{u = x^{m}} \\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x^{m-1}}\\]</p>\n    <p>\\[\\simplify[std]{v = sin({b} * x+{a})e^({n}x)}\\Rightarrow \\simplify[std]{Diff(v,x,1) = {b} * cos({b} * x+{a})e^({n}x)+{n}sin({b}x+{a})e^({n}x)}\\]</p>\n    <p>Hence on substituting into the product rule above we get:</p>\n    <p>\\[\\begin{eqnarray*}\\frac{df}{dx} &amp;=&amp; \\simplify[std]{{m}x^{m-1}sin({b} * x+{a})e^({n}x)+x^{m}({b} * cos({b} * x+{a}) * e ^ ({n} * x) + {n} * sin({b} * x+{a}) * e ^ ({n} * x))}\\\\ &amp;=&amp;\\simplify[std]{x^{m-1}e^({n}x)({b}x*cos({b}x+{a})+{n}x*sin({b}x+{a})+{m}sin({b}x+{a}))}\\\\ &amp;=&amp;\\simplify[std]{x^{m-1}e^({n}x)({b}x*cos({b}x+{a})+({n}x+{m})*sin({b}x+{a}))} \\end{eqnarray*}\\] <br/>Hence $g(x)=\\simplify[std]{{b}x*cos({b}x+{a})+({n}x+{m})*sin({b}x+{a})}$</p>\n    \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n        <p>$\\simplify[std]{f(x) = x^{m}sin({b}x + {a}) * e ^ ({n} * x)}$</p>\n        <p>The answer is of the form:</p>\n        <p>$\\displaystyle \\frac{df}{dx}= \\simplify[std]{x^{m-1}e^({n}x)g(x)}$ for a function $g(x)$.&nbsp;You have to find $g(x)$</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>if you input a function of the form $xf(x)$ where $f(x)$ is a function, then you must input it as $x*f(x)$ with * for multiplication e.g. input $x*\\sin(ax+b)$ and not $xsin(ax+b)$.</p>\n        <p>Click on Show steps for more information, you will not lose any marks by doing so.</p>\n        \n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{b}x * cos({b} * x+{a})  + ({n}x+{m}) * sin({b} * x+{a})", "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    \t\t<p><strong>31/07/2012:</strong></p>\n    \t\t    \t\t<p>Checked calculation.</p>\n    \t\t    \t\t<p>Added tags.</p>\n    \t\t    \t\t<p>Allowed no penalty on looking at Show steps.</p>\n    \t\t    \t\t<p>Corrected occurences of the form xsin and xcos to x*sin, x*cos.</p>\n    \t\t    \t\t<p>Included message warning about the input of functions of the form xsin etc.</p>\n    \t\t    \t\t<p>Show steps needs to be resolved. Now resolved.</p>\n    \t\t    \n    \t\t", "description": "\n    \t\t    \t\t<p>Differentiate $f(x)=x^{m}\\sin(ax+b) e^{nx}$.</p>\n    \t\t    \t\t<p>The answer is of the form:<br />$\\displaystyle \\frac{df}{dx}= x^{m-1}e^{nx}g(x)$ for a function $g(x)$.</p>\n    \t\t    \t\t<p>Find $g(x)$.</p>\n    \t\t    \n    \t\t", "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/"}]}