// Numbas version: finer_feedback_settings
{"name": "Differentiation:  Chain Rule - Hyp Log", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "\n\t  \n\t  \n\t  <p>Here is a table of the derivatives of some of the hyperbolic functions:</p>\n\t  \n\t  \n\t  \n\t  <table border=\"1\">\n\t  \n\t  <tr><th>$f(x)$</th><th>$\\displaystyle{\\frac{df}{dx}}$</th></tr>\n\t  \n\t  <tr><th>$\\sinh(bx)$</th><th>$b\\cosh(bx)$</th></tr>\n\t  \n\t  <tr><th>$\\cosh(bx)$</th><th>$b\\sinh(bx)$</th></tr>\n\t  \n\t  <tr><th>$\\tanh(bx)$</th><th>$\\simplify{b*sech(bx)^2}$</th></tr>\n\t  \n\t  </table>\n\t  \n\t  \n\t  \n\t  <p>a)</p>\n\t  \n\t  \n\t  \n\t  <p>$f(x)=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$</p>\n\t  \n\t  \n\t  \n\t  <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\t  \n\t  \n\t  \n\t  <p>b)</p>\n\t  \n\t  \n\t  \n\t  <p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n\t  \n\t  \n\t  \n\t  <p>Using the table above we get:<br/>\\[\\frac{df}{dx} = \\simplify[std]{{a}*sech({a}x+{b})^2}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>c)</p>\n\t  \n\t  \n\t  \n\t  <p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n\t  \n\t  \n\t  \n\t  <p>Using the chain rule we find:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\frac{df}{dx} = \\simplify[std]{{a2} * tanh({a2} * x + {b2})}\\]</p>\n\t  \n\t  \n\t    \n\t", "extensions": [], "variables": {"a2": {"group": "Ungrouped variables", "definition": "random(2..9)", "name": "a2", "templateType": "anything", "description": ""}, "b2": {"group": "Ungrouped variables", "definition": "random(-9..9)", "name": "b2", "templateType": "anything", "description": ""}, "n": {"group": "Ungrouped variables", "definition": "random(3..7)", "name": "n", "templateType": "anything", "description": ""}, "b1": {"group": "Ungrouped variables", "definition": "random(1..9)", "name": "b1", "templateType": "anything", "description": ""}, "b": {"group": "Ungrouped variables", "definition": "random(-9..9)", "name": "b", "templateType": "anything", "description": ""}, "a1": {"group": "Ungrouped variables", "definition": "random(-9..-1)", "name": "a1", "templateType": "anything", "description": ""}, "a": {"group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "templateType": "anything", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Differentiate the following functions: $\\displaystyle x ^ n \\sinh(ax + b),\\;\\tanh(cx+d),\\;\\ln(\\cosh(px+q))$</p>"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "ungrouped_variables": ["a", "b", "n", "a1", "a2", "b1", "b2"], "preamble": {"js": "", "css": ""}, "variable_groups": [], "statement": "<p></p>\n<p>Find the <strong>first derivative</strong> of each of the following functions:</p>\n<p>[NOTE: To&nbsp;input the <em>square&nbsp;</em> of a function such as $\\sinh(x)$ you have to input it as (sinh(x))^2,&nbsp; &nbsp;similarly for the other hyperbolic functions.</p>\n<p></p>", "functions": {}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"showFeedbackIcon": true, "prompt": "<p>$f(x)=\\tanh(\\simplify[std]{{a}x+{b}})$</p>\n<p>$\\displaystyle{f'(x)=\\;\\;}$[[0]]</p>", "marks": 0, "customMarkingAlgorithm": "", "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "gaps": [{"showFeedbackIcon": true, "checkVariableNames": false, "marks": 1, "customMarkingAlgorithm": "", "unitTests": [], "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showPreview": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "showCorrectAnswer": true, "answerSimplification": "std", "vsetRange": [0, 1], "checkingType": "absdiff", "variableReplacements": [], "expectedVariableNames": [], "scripts": {}, "failureRate": 1, "answer": "{a}*sech({a}x+{b})^2"}]}, {"showFeedbackIcon": true, "prompt": "<p>$y=\\simplify[std]{ x ^ {n} * sinh({b1}+{a1} * x )}$</p>\n<p>$\\displaystyle{\\frac{dy}{dx}=\\;\\;}$[[0]]</p>", "marks": 0, "customMarkingAlgorithm": "", "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "gaps": [{"showFeedbackIcon": true, "checkVariableNames": false, "marks": "2", "customMarkingAlgorithm": "", "unitTests": [], "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showPreview": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "showCorrectAnswer": true, "answerSimplification": "std", "vsetRange": [0, 1], "checkingType": "absdiff", "variableReplacements": [], "expectedVariableNames": [], "scripts": {}, "failureRate": 1, "answer": "{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})"}]}, {"showFeedbackIcon": true, "prompt": "\n\t\t\t        <p>$f(x)=\\ln(\\cosh(\\simplify[std]{{a2}x+{b2}}))$</p>\n\t\t\t        <p>$\\displaystyle{\\frac{df}{dx}=\\;\\;}$[[0]]</p>\n\t\t\t        \n\t\t\t", "marks": 0, "customMarkingAlgorithm": "", "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "gaps": [{"showFeedbackIcon": true, "checkVariableNames": false, "marks": "2", "customMarkingAlgorithm": "", "unitTests": [], "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "showPreview": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "showCorrectAnswer": true, "answerSimplification": "std", "vsetRange": [0, 1], "checkingType": "absdiff", "variableReplacements": [], "expectedVariableNames": [], "scripts": {}, "failureRate": 1, "answer": "{a2} * tanh({a2} * x + {b2})"}]}], "name": "Differentiation:  Chain Rule - Hyp Log", "type": "question", "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}