// Numbas version: finer_feedback_settings
{"name": "Differentiation: Quotient Rule 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Quotient Rule 4", "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $\\frac{e^{kx}}{ax^2+bx+c}$ using the quotient rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of \\[ \\simplify{y=e^({d}x)/({a}x^2+{b}x+{c})}. \\]</p>", "advice": "<p>If we have a function of the form $y=\\tfrac{u(x)}{v(x)}$, to calculate its derivative we need to use the quotient rule:</p>\n<p>\\[ \\dfrac{dy}{dx} = \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2}\\,.\\]</p>\n<p>This can be split up into steps:</p>\n<ol>\n<li>Identify the functions $u(x)$ and $v(x)$;</li>\n<li>Calculate their derivatives $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$;</li>\n<li>Substitute these into the formula for the quotient rule to obtain an expression for $\\tfrac{dy}{dx}$;</li>\n<li>Simplify $\\tfrac{dy}{dx}$ where possible.</li>\n</ol>\n<p></p>\n<p>Following this process, we must first identify $u(x)$ and $v(x)$.</p>\n<p>As \\[ \\simplify{y=e^({d}x)/({a}x^2+{b}x+{c})}, \\]</p>\n<p>let \\[ u(x) = \\simplify{e^({d}x)} \\quad \\text{and} \\quad v(x)=\\simplify{{a}x^2+{b}x+{c}}.\\]</p>\n<p>Next, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:</p>\n<p>\\[ \\dfrac{du}{dx} = \\simplify{{d}e^({d}x)}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=\\simplify{{2*a}x+{b}}.\\]</p>\n<p>Substituting these results into the quotient rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:</p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,= \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2} \\\\ \\\\&amp;\\,=\\dfrac{(\\simplify{{a}x^2+{b}x+{c}}) \\times\\simplify{{d}e^({d}x)} - \\simplify{e^({d}x)} \\times (\\simplify{{2*a}x+{b}})}{\\simplify{({a}x^2+{b}x+{c})^2}}.&nbsp; \\end{split}\\]</p>\n<p></p>\n<p>{advice}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "check1": {"name": "check1", "group": "Ungrouped variables", "definition": "gcd(a*d,b*d-2*a)", "description": "", "templateType": "anything", "can_override": false}, "check2": {"name": "check2", "group": "Ungrouped variables", "definition": "gcd(a*d,c*d-b)", "description": "", "templateType": "anything", "can_override": false}, "simp": {"name": "simp", "group": "Ungrouped variables", "definition": "gcd(check1,check2)", "description": "", "templateType": "anything", "can_override": false}, "simplified": {"name": "simplified", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx}&amp;\\\\,=\\\\dfrac{\\\\simplify{{d}e^({d}x)}(\\\\simplify{{a}x^2+{b}x+{c}}) - \\\\simplify{e^({d}x)}(\\\\simplify{{2*a}x+{b}})}{\\\\simplify{({a}x^2+{b}x+{c})^2}} \\\\\\\\ \\\\\\\\ &amp;\\\\,=\\\\dfrac{\\\\simplify[all,!collectNumbers,!cancelTerms]{e^({d}x)({d*a}x^2+{d*b}x+{d*c}-{2*a}x-{b})}}{\\\\simplify{({a}x^2+{b}x+{c})^2}} \\\\\\\\ \\\\\\\\&amp;\\\\,=\\\\dfrac{\\\\simplify{e^({d}x)({d*a}x^2+{d*b}x+{d*c}-{2*a}x-{b})}}{\\\\simplify{({a}x^2+{b}x+{c})^2}}.\\\\end{split} \\\\]</p>\\n<p></p>\"", "description": "", "templateType": "long string", "can_override": false}, "simplify": {"name": "simplify", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx}&amp;\\\\,=\\\\dfrac{\\\\simplify{{d}e^({d}x)}(\\\\simplify{{a}x^2+{b}x+{c}}) - \\\\simplify{e^({d}x)}(\\\\simplify{{2*a}x+{b}})}{\\\\simplify{({a}x^2+{b}x+{c})^2}} \\\\\\\\ \\\\\\\\ &amp;\\\\,=\\\\dfrac{\\\\simplify[all,!collectNumbers,!cancelTerms]{e^({d}x)({d*a}x^2+{d*b}x+{d*c}-{2*a}x-{b})}}{\\\\simplify{({a}x^2+{b}x+{c})^2}} \\\\\\\\ \\\\\\\\&amp;\\\\,=\\\\dfrac{\\\\simplify{e^({d}x)({d*a}x^2+{d*b}x+{d*c}-{2*a}x-{b})}}{\\\\simplify{({a}x^2+{b}x+{c})^2}} \\\\\\\\ \\\\\\\\&amp;\\\\,=\\\\dfrac{\\\\simplify{{simp}e^({d}x)({d*a/simp}x^2+{d*b/simp}x+{d*c/simp}-{2*a/simp}x-{b/simp})}}{\\\\simplify{({a}x^2+{b}x+{c})^2}}.\\\\end{split} \\\\]</p>\\n<p></p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(simp=1,'{simplified}','{simplify}')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "check1", "check2", "simp", "simplified", "simplify", "advice"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\dfrac{dy}{dx}=$[[0]]&nbsp;</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{simp}(e^({d}x)({a*d/simp}x^2+{(b*d-2*a)/simp}x+{(c*d-b)/simp}))/({a}x^2+{b}x+{c})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}