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{"name": "Differentiation: Product Rule 7", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Product Rule 7", "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $ax^n(bx^m+c)^p$ using the product rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of \\[ \\simplify{y={a}x^{n}({b}x^{m}+{c})^{p}}. \\]</p>", "advice": "<p>If we have a function of the form $y=u(x)v(x)$, to calculate its derivative we need to use the product rule:</p>\n<p>\\[ \\dfrac{dy}{dx} = u(x) \\times \\dfrac{dv}{dx} + v(x) \\times\\dfrac{du}{dx}.\\]</p>\n<p>This can be split up into steps:</p>\n<ol>\n<li>Identify the functions $u(x)$ and $v(x)$;</li>\n<li>Calculate their derivatives $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$;</li>\n<li>Substitute these into the formula for the product rule to obtain an expression for $\\tfrac{dy}{dx}$;</li>\n<li>Simplify $\\tfrac{dy}{dx}$ where possible.</li>\n</ol>\n<p>Following this process, we must first identify $u(x)$ and $v(x)$.</p>\n<p>As \\[ \\simplify{y={a}x^{n}({b}x^{m}+{c})^{p}}, \\]</p>\n<p>let \\[ u(x) = \\simplify{{a}x^{n}} \\quad \\text{and} \\quad v(x)=\\simplify{({b}x^{m}+{c})^{p}}.\\]</p>\n<p>Next, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:</p>\n<p>\\[ \\dfrac{du}{dx} = \\simplify{{a*n}x^{n-1}}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=\\simplify{{b*p*m}x^{m-1}({b}x^{m}+{c})^{p-1}}.\\]</p>\n<p>Substituting these results into the product rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:</p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,= \\dfrac{du}{dx}\\times v(x) + u(x) \\times\\dfrac{dv}{dx} \\\\ \\\\&amp;\\,=\\simplify{{a*n}x^{n-1}} \\times\\simplify{({b}x^{m}+{c})^{p}} +\\simplify{{a}x^{n}} \\times \\simplify{{b*p*m}x^{m-1}({b}x^{m}+{c})^{p-1}}.&nbsp; \\end{split}\\]</p>\n<p></p>\n<p>{checker}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "simp": {"name": "simp", "group": "Ungrouped variables", "definition": "gcd(n*b+b*p*m,n*c)", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(simp=1,'{simplified}','{simplify}')", "description": "", "templateType": "anything", "can_override": false}, "simplify": {"name": "simplify", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} &amp;\\\\,= \\\\simplify{{a*n}x^{n-1}({b}x^{m}+{c})^{p}+{a}x^{n}{b*p*m}x^{m-1}({b}x^{m}+{c})^{p-1}} \\\\\\\\ &amp;\\\\,= \\\\simplify[all,!expandBrackets,!cancelTerms]{{simp2}x^{n-1}({b}x^{m}+{c})^{p-1}}\\\\big(\\\\simplify[all,!expandBrackets,!cancelTerms]{{a*n/simp2}({b}x^{m}+{c})+{a*b*p*m/simp2}x^{m}}\\\\big)\\\\\\\\ &amp;\\\\,=\\\\simplify[all]{{a}x^{n-1}({b}x^{m}+{c})^{p-1}({n*b+b*p*m}x^{m}+{n*c})}\\\\\\\\ &amp;\\\\,=\\\\simplify{{a*simp}x^{n-1}({b}x^{m}+{c})^{p-1}({(n*b+b*p*m)/simp}x^{m}+{n*c/simp})}.\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "simplified": {"name": "simplified", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} &amp;\\\\,= \\\\simplify{{a*n}x^{n-1}({b}x^{m}+{c})^{p}+{a}x^{n}{b*p*m}x^{m-1}({b}x^{m}+{c})^{p-1}} \\\\\\\\ &amp;\\\\,= \\\\simplify{{a}x^{n-1}({b}x^{m}+{c})^{p-1}}\\\\big(\\\\simplify{{n}({b}x^{m}+{c})+{b*p*m}x^{m}}\\\\big)\\\\\\\\ &amp;\\\\,=\\\\simplify[all]{{a}x^{n-1}({b}x^{m}+{c})^{p-1}({n*b+b*p*m}x^{m}+{n*c})}.\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "simp2": {"name": "simp2", "group": "Ungrouped variables", "definition": "gcd(a*n,a*b*p*m)", "description": "", "templateType": "anything", "can_override": false}, "checker": {"name": "checker", "group": "Ungrouped variables", "definition": "if(gcd(simp,simp2)>1 and simp*simp2=a*n,'{advice2}','{checker2}')", "description": "", "templateType": "anything", "can_override": false}, "advice1": {"name": "advice1", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} &amp;\\\\,= \\\\simplify{{a*n}x^{n-1}({b}x^{m}+{c})^{p}+{a}x^{n}{b*p*m}x^{m-1}({b}x^{m}+{c})^{p-1}} \\\\\\\\ &amp;\\\\,= \\\\simplify[all,!expandBrackets,!cancelTerms]{{simp2}x^{n-1}({b}x^{m}+{c})^{p-1}}\\\\big(\\\\simplify[all,!expandBrackets,!cancelTerms]{{a*n/simp2}({b}x^{m}+{c})+{a*b*p*m/simp2}x^{m}}\\\\big)\\\\\\\\&amp;\\\\,=\\\\simplify[all]{{simp2}x^{n-1}({b}x^{m}+{c})^{p-1}({(a*n*b+a*b*p*m)/simp2}x^{m}+{a*n*c/simp2})}\\\\\\\\ &amp;\\\\,=\\\\simplify{{a*simp}x^{n-1}({b}x^{m}+{c})^{p-1}({(n*b+b*p*m)/simp}x^{m}+{n*c/simp})}.\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice2": {"name": "advice2", "group": "Ungrouped variables", "definition": "\"<p>Simplifying,</p>\\n<p>\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} &amp;\\\\,= \\\\simplify{{a*n}x^{n-1}({b}x^{m}+{c})^{p}+{a}x^{n}{b*p*m}x^{m-1}({b}x^{m}+{c})^{p-1}} \\\\\\\\ &amp;\\\\,= 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