// Numbas version: exam_results_page_options
{"name": "Maths Support: Product rule practice", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": false, "allowregen": false, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "<p>10 questions on the product rule in differentiation.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivatives", "differentiate a product", "differentiate polynomials", "differentiation", "elementary differentiation", "polynomials", "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 = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\\]</p>\n    \n    \n    \n    <p>Hence on substituting into the product rule above we get:</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}}\\]</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>$\\displaystyle \\simplify[std]{f(x) = x ^ {m} * ({a} * x+{b})^{n}}$</p>\n      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      <p>Clicking on <em>Show steps</em> gives you 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": "{m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}", "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"}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(3..9)", "name": "m"}, "n": {"definition": "random(3..9)", "name": "n"}}, "metadata": {"notes": "\n        \t\t<p><strong>31/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Added description.</p>\n        \t\t<p>Steps problem to be addressed via an issue. Now resolved.</p>\n        \t\t<p>Checked calculation.OK.</p>\n        \t\t<p>Improved prompt display.</p>\n        \t\t<p>Clicking on Show steps does not lose any marks.</p>\n        \t\t", "description": "<p>Differentiate $f(x) = x^m(a x+b)^n$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "derivative of a polynomial", "derivative of a product", "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 = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the product rule above we get:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}=x^{m-1}({a}x+{b})^{n-1}({m*a+n*a}x+{m*b})}\\]<br/>So $\\simplify[std]{g(x)= {m*a+n*a}x+{m*b}}$</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = x ^ {m} * ({a} * x+{b})^{n}}$</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{m*a+n*a}x+{m*b}", "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"}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule. The answer will be of the form <br/>\\[\\simplify[std]{x^{m-1}({a}x+{b})^{n-1}g(x)}\\] for a polynomial $g(x)$.&nbsp;You have to find $g(x)$</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(3..9)", "name": "m"}, "n": {"definition": "random(3..9)", "name": "n"}}, "metadata": {"notes": "\n    \t\t<p><strong>31/07/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Problem with steps to be addressed. Now resolved.</p>\n    \t\t<p>Clicking on Show steps does not lose any marks.</p>\n    \t\t", "description": "\n    \t\t<p>Differentiate $ x ^ m(ax+b)^n$&nbsp;using the product rule. The answer will be of the form&nbsp;$x^{m-1}(ax+b)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.</p>\n    \t\t<p>&nbsp;</p>\n    \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivative of a product", "differentiation", "polynomials", "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 = ({a}x+{b}) ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m*a}({a}x+{b}) ^ {m -1}}\\]</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{v = ({c} * x+{d})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*c} * ({c} * x+{d})^{n-1}}\\]</p>\n    \n    \n    \n    <p>Hence on substituting into the product rule above we get:</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{Diff(f,x,1) = {m*a}({a}x+{b}) ^ {m-1} * ({c} * x+{d})^{n}+{n*c}({a}x+{b})^{m} * ({c} * x+{d})^{n-1}=({a}x+{b})^{m-1}({c}x+{d})^{n-1}({m*a*c+n*c*a}x+{m*a*d+n*c*b})}\\]<br/>on taking out the common terms $\\simplify[std]{({a}x+{b})^{m-1}({c}x+{d})^{n-1}}$.<br/>So $\\simplify[std]{g(x)= {m*a*c+n*c*a}x+{m*a*d+n*c*b}}$</p>\n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = ({a} * x+{b})^{m}*({c}x+{d})^{n}}$</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{m*a*c+n*c*a}x+{m*a*d+n*c*b}", "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"}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule. The answer will be of the form <br/>\\[\\simplify[std]{({a}x+{b})^{m-1}({c}x+{d})^{n-1}g(x)}\\] for a polynomial $g(x)$.&nbsp;You have to find $g(x)$</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "if(a*d=b*c1,c1+1,c1)", "name": "c"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "d": {"definition": "s2*random(1..9)", "name": "d"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "random(3..9)", "name": "m"}, "n": {"definition": "random(3..9)", "name": "n"}, "c1": {"definition": "random(1..8)", "name": "c1"}}, "metadata": {"notes": "\n    \t\t<p><strong>31/07/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Corrected mistake in statement.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Problem with steps to be addressed. Now resolved.</p>\n    \t\t<p>Clicking on Show steps does not lose any marks.</p>\n    \t\t", "description": "<p>Differentiate $ (ax+b)^m(cx+d)^n$&nbsp;using the product rule. The answer will be of the form&nbsp;$(ax+b)^{m-1}(cx+d)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "differentiating a product", "differentiating trigonometric functions", "differentiation", "product rule", "steps", "trigonometric functions"], "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 = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = sin({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * cos({n} * x)}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the product rule above we get:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{Diff(f,x,1) = {m*b}({a} + {b} * x) ^ {m-1} * sin({n} * x)+{n}*({a} + {b} * x) ^ {m} * cos({n} * x)}\\]</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = ({a} + {b} * x) ^ {m} * sin({n} * x)}$</p>\n      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{m*b}({a} + {b} * x) ^ {m-1} * sin({n} * x)+{n}*({a} + {b} * x) ^ {m} * cos({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"}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..4)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..5)", "name": "b"}, "m": {"definition": "random(2..8)", "name": "m"}, "n": {"definition": "random(2..6)", "name": "n"}}, "metadata": {"notes": "\n    \t\t<p><strong>31/07/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Steps problem to be addressed. Now resolved.</p>\n    \t\t<p>Checked calculation.OK.</p>\n    \t\t<p>Improved prompt display.</p>\n    \t\t<p>Clicking on Show steps not lose any marks.</p>\n    \t\t", "description": "<p>Differentiate $ (a+bx) ^ {m} \\sin(nx)$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivative of a product", "differentiating a product of functions", "differentiating the exponential function", "differentiation", "exponential function", "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 = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]</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>\\[\\simplify[std]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) }\\]</p>\n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n        <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * 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"}], "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>Added tags.</p>\n    \t\t<p>Improved display of prompt.</p>\n    \t\t<p>Checked calculation.</p>\n    \t\t<p>Allowed no penalty on looking at Steps.</p>\n    \t\t<p>Issue with Show steps to be resolved. Has been resolved.</p>\n    \t\t", "description": "<p>Differentiate the function $(a + b x)^m &nbsp;e ^ {n x}$&nbsp;using the product rule.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "derivative of a product", "differentiating a product", "differentiating trigonometric functions", "differentiation", "product rule", "steps", "trigonometric functions"], "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 = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{v = cos({a} * x+{b})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = -{a} * sin({a} * x+{b})}\\]</p>\n    \n    \n    \n    <p>Hence on substituting into the product rule above we get:</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * cos({a} * x+{b})-{a}x^{m} * sin({a} * x+{b})}\\]</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>$\\simplify[std]{f(x) = x ^ {m} * cos({a} * x+{b})}$</p>\n      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      <p>Clicking on <em>Show steps</em> gives you 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": "{m}x ^ {m-1} * cos({a} * x+{b})-{a}x^{m} * sin({a} * x+{b})", "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"}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(2..9)", "name": "m"}}, "metadata": {"notes": "\n        \t\t<p><strong>31/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Added description.</p>\n        \t\t<p>Steps problem to be addressed. Now resolved.</p>\n        \t\t<p>Checked calculation.OK.</p>\n        \t\t<p>Improved prompt display.</p>\n        \t\t<p>Clicking on Show steps does not lose any marks.</p>\n        \t\t", "description": "<p>Differentiate $x^m\\cos(ax+b)$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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"}], "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": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "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"}], "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": []}]}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivatives", "differentiating a product", "differentiating square roots", "differentiation", "elementary 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 = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n    \n    \n    \n    <p>\\[\\simplify[std]{v = sqrt({a} * x+{b})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a}/2* ({a} * x+{b})^{-1/2}}\\]</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]{{m}x ^ {m-1} * sqrt({a} * x+{b})+{a/2}x^{m} * ({a} * x+{b})^{-1/2}}\\\\\n    \n    &#38;=&#38; \\simplify[std]{{m}x ^ {m-1} * sqrt({a} * x+{b})+{a}x^{m}/(2*sqrt({a} * x+{b}))}\\\\\n    \n    &#38;=&#38; \\simplify[std]{(2*{m}x^{m-1}({a}x+{b})+ {a}x^{m})/(2*sqrt({a} * x+{b}))}\\\\\n    \n    &#38;=&#38;\\simplify[std]{(x^{m-1}({2*m}({a}x+{b})+{a}x))/(2*sqrt({a} * x+{b}))}\\\\\n    \n    &#38;=&#38;\\simplify[std]{x^{m-1}/(2*sqrt({a} * x+{b}))({2*m*a+a}x+{2*m*b})}\n    \n    \\end{eqnarray*}\\]<br/>Hence $g(x)=\\simplify[std]{{2*m*a+a}x+{2*m*b}}$</p>\n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = x ^ {m} * sqrt({a} * x+{b})}$</p>\n        <p>The answer is in the form \\[\\frac{df}{dx}=\\simplify[std]{ x^{m-1}/(2*sqrt({a}x+{b}))g(x)}\\]<br/>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{2*m*a+a}x+{2*m*b}", "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"}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": 1.0, "name": "a"}, "b": {"definition": "random(1..9)", "name": "b"}, "m": {"definition": "random(2..9)", "name": "m"}}, "metadata": {"notes": "\n    \t\t<p><strong>31/07/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Steps problem to be addressed. Now resolved.</p>\n    \t\t<p>Checked calculation.OK.</p>\n    \t\t<p>Improved prompt display.</p>\n    \t\t<p>Clicking on Show steps does not lose any marks.</p>\n    \t\t", "description": "<p>Differentiate $ x ^m \\sqrt{a x+b}$.<br />The answer is in the form $\\displaystyle \\frac{x^{m-1}g(x)}{2\\sqrt{ax+b}}$<br />for a polynomial $g(x)$. Find $g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Product Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Steps", "algebraic manipulation", "derivative", "deriving a function", "differentiate", "differentiating a function", "differentiating a product of functions", "differentiation", "exponential function", "functions", "product rule", "steps"], "advice": "\n    \n    \n    <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{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[dPoly]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[dPoly]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]</p>\n    \n    \n    \n    <p>\\[\\simplify{v = e ^ ({n} * x)} \\Rightarrow \\simplify{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>\\[\\simplify[dPoly]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) = ({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}\\]</p>\n    \n    \n    \n    <p>The last step was to take out the common term $\\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$.</p>\n    \n    \n    \n    <p>Hence \\[\\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}\\].</p>\n    \n    \n    \n  ", "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[dPoly]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$</p>\n        <p>You are given that \\[\\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}\\]</p>\n        <p>for a polynomial $g(x)$. You have to find $g(x)$.</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>Clicking on Show steps gives you more information, you will not lose any marks by doing so.</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "dPoly", "marks": 3.0, "answer": "({((m * b) + (n * a))} + ({(n * b)} * x))", "type": "jme"}], "steps": [{"prompt": "<p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify{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"}], "statement": "<p>Differentiate the following function $f(x)$ using the product rule.</p>", "variable_groups": [], "progress": "testing", "type": "question", "variables": {"a": {"definition": "random(1..4)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..5)", "name": "b"}, "m": {"definition": "random(2..8)", "name": "m"}, "n": {"definition": "random(2..6)", "name": "n"}}, "metadata": {"notes": "\n    \t\t<p><strong>20/06/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p><strong>4/7/2012:<br /></strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p><strong>31/07/2012:</strong></p>\n    \t\t<p>Checked calculation.</p>\n    \t\t<p>Allowed no penalty on looking at Steps.</p>\n    \t\t", "description": "<p>Differentiate the function $f(x)=(a + b x)^m &nbsp;e ^ {n x}$&nbsp;using the product rule. Find $g(x)$ such that $f\\;'(x)= (a + b x)^{m-1} &nbsp;e ^ {n x}g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}