// Numbas version: exam_results_page_options
{"name": "Basic differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["g", "bc", "cc", "dc", "ec", "fc", "ap", "bp", "cp", "dp", "h", "k", "l"], "name": "Basic differentiation", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p style=\"text-align: left;\"><strong>Parts A and B</strong></p>\n<p style=\"text-align: left;\">If $y=ax^n$,</p>\n<p style=\"text-align: left;\">$\\frac{dy}{dx}=anx^{n-1}$ for all rational $n$.</p>\n<p style=\"text-align: left;\">All we have to do to terms where $x$ is to a power of anything is times the coefficient of $x$ by the original power, and then take one away from the original power.</p>\n<p style=\"text-align: left;\">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 style=\"text-align: left;\"><strong>Part C</strong></p>\n<p style=\"text-align: left;\">Remember&nbsp;that $\\ln(x)$ differentiates to $\\displaystyle \\frac {1}{x}$. It follows that $a \\ln(x)$ differentiates to&nbsp;$\\displaystyle \\frac {a}{x}$. If the $x$ term is actually, for example, $x+4$ or $2x-1$, then we also need to use the chain rule and multiply our answer by the derivative of the $x$ term.&nbsp;</p>\n<p style=\"text-align: left;\">Let's differentiate $3 \\ln(2x-1)$ as an example. Using the fact that&nbsp;$a \\ln(x)$ differentiates to $\\displaystyle \\frac {a}{x}$, we know that the first part of our answer will be</p>\n<p style=\"text-align: left;\">$\\displaystyle \\frac {3}{2x-1}$</p>\n<p style=\"text-align: left;\">We then need to find the derivative of $2x-1$ and then multiply the first part of our answer by this. The derivative of&nbsp;$2x-1$ is $2$ so the final answer will be</p>\n<p style=\"text-align: left;\">$\\displaystyle \\frac {3*2}{2x-1}$ =&nbsp;$\\displaystyle \\frac {6}{2x-1}$</p>\n<p style=\"text-align: left;\"><strong>Part D</strong></p>\n<p style=\"text-align: left;\">Remember&nbsp;that $\\cos(x)$ differentiates to $-\\sin(x)$. It follows that $a \\cos(x)$ differentiates to&nbsp;$-a \\sin(x)$. Like in the previous example, if the $x$ term is actually something slightly more complicated then we also need to use the chain rule and multiply our answer by the derivative of the $x$ term.&nbsp;</p>\n<p style=\"text-align: left;\">Let's differentiate $4 \\cos(3x+1)$ as an example. Using the fact that&nbsp;$a \\cos(x)$ differentiates to $-a \\sin(x)$, we know that the first part of our answer will be</p>\n<p style=\"text-align: left;\">$\\displaystyle -4 \\sin(3x+1)$</p>\n<p style=\"text-align: left;\">We then need to find the derivative of $3x+1$ and then multiply the first part of our answer by this. The derivative of&nbsp;$3x+1$ is $3$ so the final answer will be</p>\n<p style=\"text-align: left;\">$\\displaystyle -4*3 \\sin(3x+1)$ =&nbsp;$\\displaystyle -12 \\sin(3x+1)$</p>", "rulesets": {}, "parts": [{"prompt": "<p>$y=\\simplify[basic,zeroterm,zerofactor,unitfactor]{{dc[2]}x^{dp}+{ec[2]}x+{fc[2]}}$</p>\n<p>$\\displaystyle \\frac{dy}{dx}=$ [[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{dc[2]}*{dp}*x^{dp-1}+{ec[2]}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=\\simplify[basic,zeroterm,zerofactor,unitfactor]{{dc[3]}x^{dp}+{ec[3]}x+{fc[3]}}$</p>\n<p>$\\displaystyle \\frac{dy}{dx}=$ [[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{dc[3]}*{dp}*x^{dp-1}+{ec[3]}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=\\var{h} \\ln(x+\\var{g})$</p>\n<p>$\\displaystyle \\frac{dy}{dx}=$ [[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{h}/(x+{g})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=\\var{k} \\cos(\\var{g}x+\\var{l})$</p>\n<p>$\\displaystyle \\frac{dy}{dx}=$ [[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "-{k}*{g}*sin({g}*x+{l})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Differentiate the following functions.</p>\n<p></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"g": {"definition": "random(1..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "g", "description": ""}, "bc": {"definition": "repeat(random(-6..6),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "bc", "description": ""}, "cc": {"definition": "repeat(random(-10..10 except 0 except 1 except -1),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc", "description": ""}, "h": {"definition": "random(1..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "h", "description": ""}, "k": {"definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "k", "description": ""}, "dc": {"definition": "repeat(random(-15..15),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "dc", "description": ""}, "ec": {"definition": "repeat(random(-30..30),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "ec", "description": ""}, "l": {"definition": "random(0..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "l", "description": ""}, "ap": {"definition": "5", "templateType": "anything", "group": "Ungrouped variables", "name": "ap", "description": ""}, "fc": {"definition": "repeat(random(-50..50),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "fc", "description": ""}, "bp": {"definition": "4", "templateType": "anything", "group": "Ungrouped variables", "name": "bp", "description": ""}, "cp": {"definition": "3", "templateType": "anything", "group": "Ungrouped variables", "name": "cp", "description": ""}, "dp": {"definition": "2", "templateType": "anything", "group": "Ungrouped variables", "name": "dp", "description": ""}}, "metadata": {"description": "<p>Differentiating quadratics, natural logs and trig.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}]}], "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}