// Numbas version: exam_results_page_options
{"name": "Musa's copy of 3 Differentiation - Exponentials and Logs", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Musa's copy of 3 Differentiation - Exponentials and Logs", "tags": [], "metadata": {"description": "<p>Differentiating exponentials and Logs<br></p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Differentiate the following.</strong></p>\n<p>Do not write out $dy/dx$; only input the differentiated right hand side of each equation.</p>\n<p>Remember to enclose all single powers inside a bracket, for example, $e^{2x}$ is inputted as $e$^$(2x)$, or use $\\ln(2)$ instead of $\\ln2.$</p>", "advice": "<p><strong>The key fact to understand here is that the differentiate of $e^x$ is $e^x$.</strong></p>\n<p>This can be proven by looking at evaluating limits etc. but it is not necessary to do so at this stage.</p>\n<p>The basic steps to differentiate an exponential function are:</p>\n<p>Differentiate the power of $e$, for example in Part b,&nbsp;<strong></strong>$y=\\var{c[1]}e^{\\var{p[1]}x}$, you would differentiate $\\var{p[1]}x$.</p>\n<p>In this example, it is $\\var{p[1]}$.</p>\n<p>Then multiply the coefficient of $e$ by this result.</p>\n<p>Here, you would find $\\simplify{{c[1]}{p[1]}e^({p[1]}x)}$.</p>\n<p>This is your final answer for the derivative.</p>\n<p></p>\n<p>Remember, don't be confused if there is no coefficient. The fact the term is there means the coefficient must be $1$, but we don't tend to write it out as, for example $1x$, we just say $x$.</p>\n<p></p>\n<p>Basic formulas:</p>\n<p>$\\frac{d}{dx} e^x = e^x$</p>\n<p>$\\frac{d}{dx} e^{u(x)} = e^{u(x)}\\frac{d}{dx} u$</p>\n<p>$\\frac{d}{dx} a^x = a^x \\ln(a)$</p>\n<p>$\\frac{d}{dx} a^{u(x)} = a^{u(x)} \\ln(a) \\frac{d}{dx} u$</p>\n<p>$\\frac{d}{dx} \\ln(x) = \\frac{1}{x} ~~ (x&gt;0)$</p>\n<p>$\\frac{d}{dx} \\ln|x| = \\frac{1}{x} ~~ (x\\neq 0)$</p>\n<p>$\\frac{d}{dx} \\log_a(x) = \\frac{1}{x \\ln a} ~~ (a&gt;0, a \\neq 1)$</p>\n<p>$\\frac{d}{dx} x^x = x^x (1+\\ln x)$</p>", "rulesets": {}, "extensions": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "repeat(random(2..4),7)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..8),7)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "repeat(random(2..6),7)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "p", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$y=\\var{c[1]}e^{\\var{p[1]}x}$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({c[1]}*{p[1]})*e^({p[1]}x)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}]}