// Numbas version: finer_feedback_settings
{"name": "Hyperbolic Functions 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "n", "a1", "a2", "b1", "b2"], "name": "Hyperbolic Functions 2", "tags": ["Calculus", "Differentiation", "calculus", "chain rule", "cosh", "derivatives of hyperbolic functions", "differential", "differentiate", "differentiating hyperbolic functions", "differentiation", "hyperbolic functions", "product rule", "sinh", "tanh"], "preamble": {"css": "", "js": ""}, "advice": "<p>Here is a table of the derivatives of some of the hyperbolic functions:</p>\n<table border=\"1\">\n  <thead>\n    <tr><th>$f(x)$</th><th>$\\displaystyle{\\frac{df}{dx}}$</th></tr>\n  </thead>\n  <tbody>\n    <tr>\n      <td>$\\sinh(bx)$</td>\n      <td>$b\\cosh(bx)$</td>\n    </tr>\n    <tr>\n      <td>$\\cosh(bx)$</td>\n      <td>$b\\sinh(bx)$</td>\n    </tr>\n    <tr>\n      <td>$\\tanh(bx)$</td>\n      <td>$\\simplify{b*sech(bx)^2}$</td>\n    </tr>\n  </tbody>\n</table>\n<h4>a)</h4>\n<p>$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$</p>\n<p>Use the product rule to obtain:<br/>\\[\\frac{df}{dx} = \\simplify[std]{{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})}\\]</p>\n<h4>b)</h4>\n<p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n<p>Using the table above we get:<br/>\\[\\frac{df}{dx} = \\simplify[std]{{a}*sech({a}x+{b})^2}\\]</p>\n<h4>c)</h4>\n<p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n<p>Using the chain rule we find:</p>\n<p>\\[\\frac{df}{dx} = \\simplify[std]{{a2} * tanh({a2} * x + {b2})}\\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        <p>$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n        ", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n        <p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n        ", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a}*sech({a}x+{b})^2", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n        <p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n        ", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a2} * tanh({a2} * x + {b2})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Write down the derivatives of the following functions $f(x)$ .</p>\n<p>Note that in order to input the square of a function such as $\\sinh(x)$ you have to input it as <code>sinh(x))^2</code>, similarly for the other hyperbolic functions.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "n": {"definition": "random(3..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "a1": {"definition": "random(-9..-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "b1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(-9..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>29/06/2012:</strong></p>\n        \t\t    <p>Added and edited tags.</p>\n        \t\t    <p><strong>19/07/2012:</strong></p>\n        \t\t    <p>Added description.</p>\n        \t\t    <p>There is also the problem of inputting functions of the form $xf(x)$ if $n=1$ or $2$ in the first question. So have reset $n$ to between $3$ and $7$. Otherwise would have to have an instruction here (perhaps depending on value of $n$).</p>\n        \t\t    <p>Checked calculation.</p>\n        \t\t    <p><strong>23/07/2012:</strong></p>\n        \t\t    <p>Added tags.</p>\n        \t\t    <p></p>\n        \t\t    <p>Question appears to be working correctly.</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t", "description": "<p>Differentiate the following functions: $\\displaystyle x ^ n \\sinh(ax + b),\\;\\tanh(cx+d),\\;\\ln(\\cosh(px+q))$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}