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{"name": "Quotient Rule 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p><strong>These questions use the&nbsp;quotient&nbsp;rule.</strong></p>\n<p>The quotient&nbsp;rule is defined as</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=\\frac{v\\frac{du}{dx}+u\\frac{dv}{dx}}{v^2}$</p>\n<p style=\"text-align: center;\">when $y=\\frac{u}{v}$</p>\n<p><strong>Worked example using Part a:</strong></p>\n<p style=\"text-align: center;\"><fraction><numerator>$x$</numerator><denominator>$\\simplify{x+{c[0]}}$</denominator></fraction></p>\n<p style=\"text-align: left;\">This expression is the result of&nbsp;$x$ divided by&nbsp;($\\simplify{x+{c[0]}}$).</p>\n<p style=\"text-align: left;\">We can therefore say:</p>\n<p style=\"text-align: left;\">$u=x$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$v=\\simplify{x+{c[0]}}$,</p>\n<p style=\"text-align: left;\">Hence meaning that $y=\\frac{u}{v}$.</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 quotient&nbsp;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;\">$\\frac{dy}{dx}=\\frac{1(\\simplify{x+{c[0]}})+1(x)}{\\simplify{(x+{c[0]})^2}}$</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;\">$\\frac{dy}{dx}=\\simplify{((2x+{c[0]}))/(x+{c[0]})^2}$</p>", "variable_groups": [], "preamble": {"css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "extensions": [], "variables": {"p": {"definition": "repeat(random(2..7),6)", "templateType": "anything", "name": "p", "group": "Ungrouped variables", "description": ""}, "c": {"definition": "repeat(random(-9..9 except 0),12)", "templateType": "anything", "name": "c", "group": "Ungrouped variables", "description": ""}}, "statement": "<p><strong>Differentiate the following expressions with respect to $x$ using the quotient rule.</strong></p>\n<p><strong></strong>Simplify your answers as much as possible.</p>", "tags": [], "functions": {}, "name": "Quotient Rule 3", "parts": [{"vsetrangepoints": 5, "scripts": {}, "answersimplification": "all", "checkvariablenames": true, "answer": "({c[1]}*{c[3]}-{c[2]})/(x+{c[3]})^2", "showCorrectAnswer": false, "checkingaccuracy": 0.001, "variableReplacements": [], "type": "jme", "prompt": "<p><fraction><numerator>$\\simplify{{c[1]}x+{c[2]}}$</numerator><denominator>$\\simplify{x+{c[3]}}$</denominator></fraction></p>", "marks": "3", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showpreview": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "expectedvariablenames": ["x"]}, {"vsetrangepoints": 5, "scripts": {}, "answersimplification": "all", "checkvariablenames": true, "answer": "({c[5]}*{p[0]}*x^({p[0]}-1)+({c[4]}*{p[0]}-{c[4]})*x^{p[0]})/({c[4]}x+{c[5]})^2", "showCorrectAnswer": false, "checkingaccuracy": 0.001, "variableReplacements": [], "type": "jme", "prompt": "<p><fraction><numerator>$\\simplify{x^{p[0]}}$</numerator><denominator>$\\simplify{{c[4]}x+{c[5]}}$</denominator></fraction></p>", "marks": "4", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showpreview": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "expectedvariablenames": ["x"]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>An introduction to using the quotient rule</p>"}, "ungrouped_variables": ["c", "p"], "rulesets": {"std": ["all"]}, "type": "question", "contributors": [{"name": "Milena Venkova", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2169/"}]}]}], "contributors": [{"name": "Milena Venkova", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2169/"}]}