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{"name": "Differentiation 11 - Product Rule (Basic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "", "css": ""}, "ungrouped_variables": ["c", "p", "c1"], "advice": "<p><strong>These questions use the product rule.</strong></p>\n<p>The product rule is defined as</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=v\\frac{du}{dx}+u\\frac{dv}{dx}$</p>\n<p style=\"text-align: center;\">when $y=uv$</p>\n<p><strong>Worked example using Part a:</strong></p>\n<p style=\"text-align: center;\">$y=\\simplify{(x+{c[0]})(x+{c1})}$</p>\n<p style=\"text-align: left;\">This expression is the result of two products;&nbsp;$(\\simplify{x+{c[0]}})$ and&nbsp;$(\\simplify{x+{c1}})$.</p>\n<p style=\"text-align: left;\">We can therefore say:</p>\n<p style=\"text-align: left;\">$u=(\\simplify{x+{c[0]}})$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$v=(\\simplify{x+{c1}})$,</p>\n<p style=\"text-align: left;\">Hence meaning that $y=uv$.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We already have what $u$ and $v$ equal, so all we have to do is find what $\\frac{du}{dx}$ and $\\frac{dv}{dx}$ are, and then substitute everything into the rule.</p>\n<p style=\"text-align: left;\">Differentiating with respect to $x$, we get:</p>\n<p style=\"text-align: left;\">$\\frac{du}{dx}=1$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$\\frac{dv}{dx}=1$.</p>\n<p style=\"text-align: left;\">As there are no powers or coefficients of $x$ that are $&gt;1$, this is a very simple version of the product rule, but knowing how to work out this equation formally will make more difficult looking problems just as simple.</p>\n<p style=\"text-align: left;\">Substituting in all the results we've found, we get:</p>\n<p style=\"text-align: left;\">$\\frac{dy}{dx}=1(\\simplify{x+{c1}})+1(\\simplify{x+{c[0]}})$</p>\n<p style=\"text-align: left;\">We then simplify, collecting all the terms, to get our final answer of:</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=\\simplify{2x+{c[0]}+{c1}}$</p>", "metadata": {"description": "<p>Introduction to using the product rule</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "variablesTest": {"maxRuns": "187", "condition": "\n"}, "parts": [{"prompt": "<p>$\\simplify{(x+{c[0]})(x+{c1})}$</p>", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": "2", "checkVariableNames": false, "valuegenerators": [{"value": "", "name": "x"}], "customName": "", "answer": "2x+{c1}+{c[0]}", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}, {"prompt": "<p>$\\simplify{({c[2]}x+{c[3]})({c[4]}x+{c[5]})}$</p>", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": "2", "checkVariableNames": false, "valuegenerators": [{"value": "", "name": "x"}], "customName": "", "answer": "({c[2]}{c[4]}*2x)+({c[3]}{c[4]}+{c[2]}{c[5]})", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}, {"prompt": "<p>$\\simplify{(x^2+{c[6]})({c[7]}x+{c[8]})}$</p>", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": "2", "checkVariableNames": false, "valuegenerators": [{"value": "", "name": "x"}], "customName": "", "answer": "({c[7]}*3x^2)+(2{c[8]}x)+({c[7]}{c[6]})", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}, {"prompt": "<p>$\\simplify{(x^2+{c[9]})(x^3+{c[10]})}$</p>", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": "2", "checkVariableNames": true, "valuegenerators": [{"value": "", "name": "x"}], "customName": "", "answer": "5x^4+3{c[9]}x^2+2{c[10]}x", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}], "tags": [], "statement": "<p><strong>Differentiate the following expressions with respect to $x$ using the product rule.</strong></p>\n<p><strong></strong>Simplify your answers as much as possible.</p>", "variables": {"p": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(2..6),10)", "name": "p"}, "c1": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(-9..9 except 0 except c[0])", "name": "c1"}, "c": {"templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "repeat(random(-9..9 except 0),11)", "name": "c"}}, "rulesets": {}, "functions": {}, "extensions": [], "name": "Differentiation 11 - Product Rule (Basic)", "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}