// Numbas version: finer_feedback_settings
{"name": "Chain rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Chain rule", "tags": ["chain rule", "cosh", "derivatives of hyperbolic functions", "differential", "differentiate", "differentiating hyperbolic functions", "differentiation", "hyperbolic functions", "product rule", "sinh", "tanh"], "advice": "\n    \n    \n    <p>Here is a table of the derivatives of some of the hyperbolic functions:</p>\n    \n    \n    \n    <table border=\"1\">\n    \n    <tr><th>$f(x)$</th><th>$\\displaystyle{\\frac{df}{dx}}$</th></tr>\n    \n    <tr><th>$\\sinh(bx)$</th><th>$b\\cosh(bx)$</th></tr>\n    \n    <tr><th>$\\cosh(bx)$</th><th>$b\\sinh(bx)$</th></tr>\n    \n    <tr><th>$\\tanh(bx)$</th><th>$\\simplify{b*sech(bx)^2}$</th></tr>\n    \n    </table>\n    \n    \n    \n    <p>a)</p>\n    \n    \n    \n    <p>$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$</p>\n    \n    \n    \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    \n    \n    \n    <p>b)</p>\n    \n    \n    \n    <p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n    \n    \n    \n    <p>Using the table above we get:<br/>\\[\\frac{df}{dx} = \\simplify[std]{{a}*sech({a}x+{b})^2}\\]</p>\n    \n    \n    \n    <p>c)</p>\n    \n    \n    \n    <p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n    \n    \n    \n    <p>Using the chain rule we find:</p>\n    \n    \n    \n    <p>\\[\\frac{df}{dx} = \\simplify[std]{{a2} * tanh({a2} * x + {b2})}\\]</p>\n    \n    \n  \n  ", "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      \n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n      \n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a}*sech({a}x+{b})^2", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n      <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n      \n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a2} * tanh({a2} * x + {b2})", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n    \n    \n    <p>Write down the derivatives of the following functions $f(x)$ .</p>\n    \n    \n    \n    <p>Note that in order to input the square of a function such as $\\sinh(x)$ you have to input it as $(\\sinh(x))^2$, similarly for the other hyperbolic functions.</p>\n    \n    \n  \n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "b": {"definition": "random(-9..9)", "name": "b"}, "n": {"definition": "random(3..7)", "name": "n"}, "a1": {"definition": "random(-9..-1)", "name": "a1"}, "a2": {"definition": "random(2..9)", "name": "a2"}, "b1": {"definition": "random(1..9)", "name": "b1"}, "b2": {"definition": "random(-9..9)", "name": "b2"}}, "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>&nbsp;</p>\n    \t\t<p>Question appears to be working correctly.</p>\n    \t\t<p><strong>1/08/2012:</strong></p>\n    \t\t<p>This is a copy of MAS114220122013CBA3_4 and is included in Diagnostic: Chain Rule Practice exam.</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"}, "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/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}