// Numbas version: exam_results_page_options {"name": "Harry's copy of Differentiate product of binomial and exponential", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Harry's copy of Differentiate product of binomial and exponential", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "
Differentiate the function $(a + b x)^m e ^ {n x}$ using the product rule.
", "notes": "\n\t\t31/07/2012:
\n\t\tAdded tags.
\n\t\tImproved display of prompt.
\n\t\tChecked calculation.
\n\t\tAllowed no penalty on looking at Steps.
\n\t\tIssue with Show steps to be resolved. Has been resolved.
\n\t\t"}, "tags": ["algebraic manipulation", "Calculus", "checked2015", "derivative of a product", "differentiating a product of functions", "differentiating the exponential function", "differentiation", "exponential function", "MAS1601", "product rule", "Steps"], "advice": "\n\t \n\t \n\tThe product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
For this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = e ^ ({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]
\n\t \n\t \n\t \n\tHence on substituting into the product rule above we get:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) }\\]
\n\t \n\t \n\t", "statement": "Differentiate the following function $f(x)$ using the product rule.
", "variable_groups": [], "variables": {"a": {"templateType": "anything", "name": "a", "group": "Ungrouped variables", "definition": "random(1..9)", "description": ""}, "s2": {"templateType": "anything", "name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": ""}, "s1": {"templateType": "anything", "name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": ""}, "m": {"templateType": "anything", "name": "m", "group": "Ungrouped variables", "definition": "random(2..8)", "description": ""}, "b": {"templateType": "anything", "name": "b", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": ""}, "n": {"templateType": "anything", "name": "n", "group": "Ungrouped variables", "definition": "s2*random(2..5)", "description": ""}}, "type": "question", "showQuestionGroupNames": false, "preamble": {"js": "", "css": ""}, "rulesets": {"surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}], "std": ["all", "!collectNumbers", "fractionNumbers"]}, "ungrouped_variables": ["a", "b", "s2", "s1", "m", "n"], "parts": [{"prompt": "\n\t\t\t$\\simplify[std]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$
\n\t\t\t$\\displaystyle \\frac{df}{dx}=\\;$[[0]]
\n\t\t\tClicking on Show steps gives you more information, you will not lose any marks by doing so.
\n\t\t\t", "type": "gapfill", "showCorrectAnswer": true, "steps": [{"prompt": "The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]