// Numbas version: finer_feedback_settings {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Product rule", "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "derivatives", "differentiating a product", "differentiating square roots", "differentiation", "elementary differentiation", "product rule", "steps"], "advice": "\n \n \n

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)}\\]

\n \n \n \n

For this example:

\n \n \n \n

\\[\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]

\n \n \n \n

\\[\\simplify[std]{v = sqrt({a} * x+{b})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {a}/2* ({a} * x+{b})^{-1/2}}\\]

\n \n \n \n

Hence on substituting into the product rule above we get:

\n \n \n \n

\\[\\begin{eqnarray*} \\frac{df}{dx}&=& \\simplify[std]{{m}x ^ {m-1} * sqrt({a} * x+{b})+{a/2}x^{m} * ({a} * x+{b})^{-1/2}}\\\\\n \n &=& \\simplify[std]{{m}x ^ {m-1} * sqrt({a} * x+{b})+{a}x^{m}/(2*sqrt({a} * x+{b}))}\\\\\n \n &=& \\simplify[std]{(2*{m}x^{m-1}({a}x+{b})+ {a}x^{m})/(2*sqrt({a} * x+{b}))}\\\\\n \n &=&\\simplify[std]{(x^{m-1}({2*m}({a}x+{b})+{a}x))/(2*sqrt({a} * x+{b}))}\\\\\n \n &=&\\simplify[std]{x^{m-1}/(2*sqrt({a} * x+{b}))({2*m*a+a}x+{2*m*b})}\n \n \\end{eqnarray*}\\]
Hence $g(x)=\\simplify[std]{{2*m*a+a}x+{2*m*b}}$

\n \n \n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n

$\\simplify[std]{f(x) = x ^ {m} * sqrt({a} * x+{b})}$

\n

The answer is in the form \\[\\frac{df}{dx}=\\simplify[std]{ x^{m-1}/(2*sqrt({a}x+{b}))g(x)}\\]
for a polynomial $g(x)$. You have to find $g(x)$.

\n

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

\n

Clicking on Show steps gives you more information, you will not lose any marks by doing so.

\n ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "{2*m*a+a}x+{2*m*b}", "type": "jme"}], "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)}\\]

", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "

Differentiate the following function $f(x)$ using the product rule.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": 1.0, "name": "a"}, "b": {"definition": "random(1..9)", "name": "b"}, "m": {"definition": "random(2..9)", "name": "m"}}, "metadata": {"notes": "\n \t\t

31/07/2012:

\n \t\t

Added tags.

\n \t\t

Added description.

\n \t\t

Steps problem to be addressed. Now resolved.

\n \t\t

Checked calculation.OK.

\n \t\t

Improved prompt display.

\n \t\t

Clicking on Show steps does not lose any marks.

\n \t\t", "description": "

Differentiate $ x ^m \\sqrt{a x+b}$.
The answer is in the form $\\displaystyle \\frac{x^{m-1}g(x)}{2\\sqrt{ax+b}}$
for a polynomial $g(x)$. Find $g(x)$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}