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{"name": "Differentiation 6 - Exponentials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["c", "p"], "name": "Differentiation 6 - Exponentials", "tags": [], "preamble": {"css": "", "js": ""}, "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>", "rulesets": {}, "parts": [{"prompt": "<p>$y=e^x+\\var{c[4]}x^2+\\var{c[5]}x+\\var{c[6]}$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["e", "x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "e^x+2{c[4]}x+{c[5]}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=\\var{c[1]}e^{\\var{p[1]}x}$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["e", "x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c[1]}*{p[1]})*e^({p[1]}x)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=\\var{c[2]}e^{\\var{p[2]}x}+\\sin(x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["e", "x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c[2]}*{p[2]})*e^({p[2]}x)+cos(x)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=-\\var{c[4]}\\tan(x)+\\var{c[3]}e^{\\var{p[3]}x}-\\cos(x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["e", "x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "-{c[4]}sec^2(x)+({c[3]}*{p[3]})*e^({p[3]}x)+sin(x)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "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 $e$^$(2x)$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"p": {"definition": "repeat(random(2..4),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "c": {"definition": "repeat(random(2..8),7)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}}, "metadata": {"notes": "", "description": "<p>Differentiating exponentials</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}