Differentiate the function $s=x+x^{\\var{n}} \\cos (\\var{a}x+\\var{b})$.
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", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"x-{a}x^{n}*sin({a}x+{b})+{n}x^{n-1}*cos({a}x+{b})\",\"Almost there! Don't forget to differentiate the first term (the $x$) as well!\"],\n [\"1-{a*n}*x^{n-1}*sin({a}x+{b})\", \"Take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule.\"],\n [\"1-{n}x^{n-1}*sin({a})\", \"There are two issues here. Firstly, take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule. Secondly, remember that the angle of a trigonometric function never changes when you differentiate!\"],\n [\"-{a}(x+x^{n})*sin({a}x+{b})+(1+{n}x^{n-1})*cos({a}x+{b})\", \"Be very careful when using the product rule how you decide what $u$ and $v$ are. If $s=(x+x^{\\\\var{n}})\\\\cos (\\\\var{a}x+\\\\var{b})$, then $u=x+x^{\\\\var{n}}$ and $v=\\\\cos (\\\\var{a}x+\\\\var{b})$. However, here, $s=x+x^{\\\\var{n}} \\\\cos (\\\\var{a}x+\\\\var{b})$ (i.e. there are no brackets). Therefore, $u=x^{\\\\var{n}}$ and $v=\\\\cos (\\\\var{a}x+\\\\var{b})$. The $x$ being added at the start is differentiated separately.\"],\n [\"-{a}(1+{n}x^{n-1})*sin({a}x+{b})\", \"Take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule.\"],\n [\"-(1+{n}x^{n-1})*sin({a})\", \"Take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule. Also, remember that the angle of a trigonometric function never changes when you differentiate!\"],\n [\"1+{a}x^{n}*sin({a}x+{b})+{n}x^{n-1}*cos({a}x+{b})\", \"Almost there! Double check the rule for differentiating $\\\\cos x$.\"],\n [\"1-x^{n}*sin({a}x+{b})+{n}x^{n-1}*cos({a}x+{b})\", \"You have used the correct rule for differentiating $\\\\cos x$, but when the angle is more complicated than $x$, what extra step do you need to include?\"],\n [\"1-x^{n}*sin({a})+{n}x^{n-1}*cos({a}x+{b})\",\"Remember, the angle in a trigonometric function never changes when you differentiate!\"],\n [\"1+x^{n}*sin({a})+{n}x^{n-1}*cos({a}x+{b})\",\"Remember, the angle in a trigonometric function never changes when you differentiate! Also, double check the rule for differentiating $\\\\cos x$.\"] \n ]\n \nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))