// Numbas version: exam_results_page_options {"name": "Luis's copy of Quotient rule - differentiate linear over square root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": ["algebraic manipulation", "Calculus", "checked2015", "derivative of a quotient", "differentiation", "MAS1601", "quotient rule", "Steps"], "statement": "

Differentiate the following function $f(x)$ using the quotient rule or otherwise.

", "name": "Luis's copy of Quotient rule - differentiate linear over square root", "metadata": {"notes": "\n\t\t

1/08/2012:

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Added tags.

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Added description.

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Checked calculation. OK.

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Added information about Show steps. Altered to 0 marks lost rather than 1.

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Changed std rule set to include !noLeadingMinus, so polynomials don't change order. Got rid of a redundant ruleset.

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Improved display in various places.

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Added condition that numbers have to be input as fractions or integers - added decimal point to forbidden strings.

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The derivative of  $\\displaystyle \\frac{ax+b}{\\sqrt{cx+d}}$ is $\\displaystyle \\frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.

"}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "d1"], "parts": [{"prompt": "\n\t\t\t

\\[\\simplify[std]{f(x) = ({a} * x + {b}) / Sqrt({c} * x + {d})}\\]

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You are given that \\[\\simplify[std]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}\\]

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for a polynomial $g(x)$. You have to find $g(x)$.

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Input all numbers as fractions or integers.

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You can click on Show steps to get help. You will not lose any marks if you do so.

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$g(x)=\\;$[[0]]

\n\t\t\t", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"checkvariablenames": false, "showCorrectAnswer": true, "answer": "(({(a * c)} * x) + {((2 * a * d) + ( - (c * b)))})", "vsetrange": [0, 1], "answersimplification": "all", "expectedvariablenames": [], "vsetrangepoints": 5, "type": "jme", "scripts": {}, "checkingtype": "absdiff", "notallowed": {"partialCredit": 0, "message": "

Input all numbers as fractions or integers.

", "showStrings": false, "strings": ["."]}, "marks": 3, "checkingaccuracy": 0.001, "showpreview": true}], "stepsPenalty": 0, "scripts": {}, "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}\\]

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The quotient rule says that if $u$ and $v$ are functions of $x$ then

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\\[\\simplify[std]{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}\\]

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For this example:

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\\[\\simplify[std]{u = {a} * x + {b}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]

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\\[\\simplify[std]{v = Sqrt({c} * x + {d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}\\]

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Hence on substituting into the quotient rule above we get:

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\\[\\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))}\\]

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Hence \\[\\simplify[std]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}\\].

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