// Numbas version: exam_results_page_options
{"name": "Differentiation: First Principles 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: First Principles 4", "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $f(x)=\\frac{1}{ax}$ from first principles.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Differentiate $f(x)=\\frac{1}{\\simplify{{a}*x}}$ from first principles.</p>", "advice": "<p>For a function $f(x)$, its derivative is defined as \\[\\frac{df}{dx}=\\frac{f(x+h) - f(x)}{h} \\quad\\text{in the limit as $h$ tends to $0$.}\\]</p>\n<p>This is written \\[ \\frac{df}{dx}= \\lim_\\limits{h\\to 0}\\frac{f(x+h) - f(x)}{h}.\\]</p>\n<p>So, for the function $f(x)=\\frac{1}{\\simplify{{a}*x}}$,</p>\n<p>\\begin{split}\\frac{df}{dx}&amp;=\\lim_\\limits{h\\to 0} \\frac{f(x+h)-f(x)}{h}\\\\ &amp;=\\lim_\\limits{h\\to 0} \\frac{\\frac{1}{\\simplify{{a}(x+h)}}-\\frac{1}{\\simplify{{a}x}}}{h}\\\\ &amp;=\\lim_\\limits{h\\to 0} \\frac{\\frac{\\simplify{{a}x-({a}x+{a}h)}}{\\simplify{({a}x+{a}h)({a}x)}}}{h}\\\\ &amp;=\\lim_\\limits{h\\to 0}\\frac{\\simplify{-{a}h}}{h(\\simplify{{a}^2*x(x+h)}}\\\\ &amp;=\\lim_\\limits{h\\to 0}\\frac{\\simplify{-{a}}}{\\simplify{{a}^2*x(x+h)}}\\\\&amp;=\\lim_\\limits{h\\to 0}\\frac{-1}{\\simplify{{a}*x(x+h)}}\\\\&amp;=\\frac{-1}{\\simplify{{a}*x^2}}\\end{split}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 9#1)", "description": "<p>constant</p>", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a"], "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>$\\frac{df}{dx}= $[[0]]</p>\n<p></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": "-1/({a}*x^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", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}