The quotient rule says that if $u$ and $v$ are functions of $x$ then
\n \n \n \n\\[\\simplify[std]{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\\]
\n \n \n \nFor this example:
\n \n \n \n\\[\\simplify[std]{u = {a} * x + {b}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]
\n \n \n \n\\[\\simplify[std]{v = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}\\]
\n \n \n \nHence on substituting into the quotient rule above we get:
\n \n \n \n\\[\\simplify[std]{Diff(f,x,1) = ({a} * Sqrt({c} * x + {d}) -(({a} * x + {b}) * Diff(v,x,1))) / ({c} * x + {d}) = ({a} * Sqrt({c} * x + {d}) -(({c} * ({a} * x + {b})) / (2 * Sqrt({c} * x + {d})))) / ({c} * x + {d}) = ({2 * a} * ({c} * x + {d}) -({c} * ({a} * x + {b}))) / (2 * ({c} * x + {d}) ^ (3 / 2)) = ({a * c} * x + {2 * a * d -(c * b)}) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]
\n \n \n \nHence \\[\\simplify[std]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].
\n \n \n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n\\[\\simplify[std]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]
\nYou are given that \\[\\simplify[std]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]
\nfor a polynomial $g(x)$. You have to find $g(x)$.
\nInput all numbers as fractions or integers.
\nYou can click on Show steps to get help. You will not lose any marks if you do so.
\n$g(x)=\\;$[[0]]
\n ", "gaps": [{"notallowed": {"message": "Input all numbers as fractions or integers.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all", "marks": 3.0, "answer": "(({(a * c)} * x) + {((2 * a * d) + ( - (c * b)))})", "type": "jme"}], "steps": [{"prompt": "The quotient rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\\]
Differentiate the following function $f(x)$ using the quotient rule or otherwise.
", "variable_groups": [], "progress": "ready", "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "variables": {"a": {"definition": "random(1..8)", "name": "a"}, "c": {"definition": "random(1,3,5,7)", "name": "c"}, "b": {"definition": "if(2|a,random(-7..7#2),random(-8..8#2))", "name": "b"}, "d": {"definition": "if(a*d1=b*c,abs(d1)+1,d1)", "name": "d"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "d1": {"definition": "s1*random(1..8)", "name": "d1"}}, "showQuestionGroupNames": false, "metadata": {"notes": "\n \t\t1/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tAdded information about Show steps. Altered to 0 marks lost rather than 1.
\n \t\tChanged std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.
\n \t\tImproved display in various places.
\n \t\tAdded condition that numbers have to be input as fractions or integers - added decimal point to forbidden strings.
\n \t\t", "description": "The derivative of $\\displaystyle \\frac{ax+b}{\\sqrt{cx+d}}$ is $\\displaystyle \\frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.
", "licence": "Creative Commons Attribution 4.0 International"}}]}], "showQuestionGroupNames": false, "metadata": {"description": "8 questions on the quotient rule in differentiation.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "exam", "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}], "extensions": [], "custom_part_types": [], "resources": []}