// Numbas version: exam_results_page_options {"name": "Luis's copy of Differentation: Product Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "
Differentiate the following function $f(x)$ using the product rule.
", "tags": ["algebraic manipulation", "checked2015", "derivative", "derivative ", "deriving a function", "differentiate", "differentiating a function", "differentiating a product of functions", "differentiation", "exponential function", "functions", "mas1601", "MAS1601", "product rule", "Steps", "steps"], "functions": {}, "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "variables": {"m": {"group": "Ungrouped variables", "definition": "random(2..8)", "templateType": "anything", "description": "", "name": "m"}, "b": {"group": "Ungrouped variables", "definition": "s1*random(1..5)", "templateType": "anything", "description": "", "name": "b"}, "a": {"group": "Ungrouped variables", "definition": "random(1..4)", "templateType": "anything", "description": "", "name": "a"}, "s1": {"group": "Ungrouped variables", "definition": "random(1,-1)", "templateType": "anything", "description": "", "name": "s1"}, "n": {"group": "Ungrouped variables", "definition": "random(2..6)", "templateType": "anything", "description": "", "name": "n"}}, "name": "Luis's copy of Differentation: Product Rule", "ungrouped_variables": ["a", "s1", "b", "m", "n"], "variable_groups": [], "type": "question", "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "Differentiate the function $f(x)=(a + b x)^m e ^ {n x}$ using the product rule. Find $g(x)$ such that $f\\;'(x)= (a + b x)^{m-1} e ^ {n x}g(x)$.
", "notes": "\n\t\t20/06/2012:
\n\t\tAdded tags.
\n\t\t4/7/2012:
Added tags.
\n\t\t31/07/2012:
\n\t\tChecked calculation.
\n\t\tAllowed no penalty on looking at Steps.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "prompt": "\n\t\t\t$\\simplify[dPoly]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$
\n\t\t\tYou are given that \\[\\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}\\]
\n\t\t\tfor a polynomial $g(x)$. You have to find $g(x)$.
\n\t\t\t$g(x)=\\;$[[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", "steps": [{"type": "information", "prompt": "The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify{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[dPoly]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[dPoly]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]
\n\t \n\t \n\t \n\t\\[\\simplify{v = e ^ ({n} * x)} \\Rightarrow \\simplify{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[dPoly]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) = ({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}\\]
\n\t \n\t \n\t \n\tThe last step was to take out the common term $\\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$.
\n\t \n\t \n\t \n\tHence \\[\\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}\\].
\n\t \n\t \n\t \n\t", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}