// 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)}\\]
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 \nHence 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 $g(x)=\\simplify[std]{{2*m*a+a}x+{2*m*b}}$
$\\simplify[std]{f(x) = x ^ {m} * sqrt({a} * x+{b})}$
\nThe 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 $g(x)$.
$g(x)=\\;$[[0]]
\nClicking 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)}\\]
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\t31/07/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tSteps problem to be addressed. Now resolved.
\n \t\tChecked calculation.OK.
\n \t\tImproved prompt display.
\n \t\tClicking 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 $g(x)$.