// Numbas version: finer_feedback_settings
{"name": "Differentiate some hyperbolic functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiate some hyperbolic functions", "tags": ["Calculus", "calculus", "chain rule", "cosh", "derivatives of hyperbolic functions", "differential", "differentiate", "differentiating hyperbolic functions", "differentiation", "Differentiation", "hyperbolic functions", "product rule", "sinh", "tanh"], "metadata": {"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"}, "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>", "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"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..7)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-9..-1)", "description": 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"variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$y=\\simplify[std]{ x ^ {n} * sinh({a1} * x + {b1})}$</p>\n<p>$\\displaystyle{\\frac{dy}{dx}=\\;\\;}$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{n} * (x ^ {(n -1)}) * sinh({a1} * x + {b1}) + {a1} * (x ^ {n}) * Cosh({a1} * x + {b1})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": 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