// Numbas version: exam_results_page_options
{"name": "Power Rule 03", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Power Rule 03", "tags": [], "metadata": {"description": "<p>Simple application of \"Power Rule\" to differentiate polynomials.</p>\n<p>Some co-efficients and powers are non-integer and some may be negative.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h4>The Power Rule</h4>\n<p>You can find the derivative for powers of functions using the following rule:</p>\n<p style=\"text-align: center;\">If&nbsp; &nbsp;\\( y=ax^n \\)&nbsp; then&nbsp; &nbsp;\\( \\frac{dy}{dx} = n \\times a x^{n-1} \\)&nbsp;</p>\n<h4>The Sum or Difference Rules</h4>\n<p style=\"text-align: center;\">The derivative of&nbsp; &nbsp;\\( f(x) + g(x) \\)&nbsp; is&nbsp; \\(&nbsp; \\frac{df}{dx} + \\frac{dg}{dx} \\)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; the derivative of&nbsp; &nbsp;\\( f(x) - g(x) \\)&nbsp; is&nbsp; \\(&nbsp; \\frac{df}{dx} - \\frac{dg}{dx} \\)</p>\n<p></p>", "advice": "<p>We are asked to differentiate a variety of functions, each consisting of a single term.</p>\n<p>We can do this using the \"<em>Power Rule</em>\" for differentiation:</p>\n<p style=\"text-align: center;\">If&nbsp; &nbsp;\\( y=ax^n \\)&nbsp; then&nbsp; &nbsp;\\( \\frac{dy}{dx} = n \\times a x^{n-1} \\)</p>\n<p style=\"text-align: left;\">In plain language, \"<em>multiply by the power, then reduce the power by one</em>\".</p>\n<p style=\"text-align: left;\">The \"<em>Sum or Difference Rules</em>\" also tell us that, <strong>as long as the terms are either added or subtracted</strong>, we can differentiate the function term by term.</p>\n<p style=\"text-align: left;\">Then:</p>\n<p style=\"text-align: left;\">a)</p>\n<p style=\"padding-left: 40px;\">\\( y= \\var{a1} x^{\\var{n1}} + \\var{a2} x^{\\var{n2}} +\\var{a3} \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } =\\var{n1} \\times \\var{a1} x^{\\var{n1} - 1} + \\var{n2} \\times \\var{a2} x^{\\var{n2}-1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">The constant term&nbsp; ( \\( \\var{a3} \\) )&nbsp; can be seen as &nbsp;\\( \\var{a3}x^0 \\) so will differentiate to \\( 0 \\times \\var{a3} x^{0-1} \\) which, of course equals zero.&nbsp;</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{ {n1}*{a1}*x^({n1}-1) + {n2}*{a2}*x^({n2}-1) } \\)</p>\n<p style=\"padding-left: 40px;\"></p>\n<p style=\"text-align: left;\">b)</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{y= {b1} x^{{m1}} + {b2} x^{{m2}}+{b3}x^{{m3}}+{b4} }&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\var{m1} \\times \\var{b1} x^{\\var{m1} - 1} +\\var{m2} \\times \\var{b2} x^{\\var{m2} - 1}+\\var{m3} \\times \\var{b3} x^{\\var{m3} - 1}\\)</p>\n<p style=\"padding-left: 40px;\">The constant term&nbsp; ( \\( \\var{b4} \\) )&nbsp; can be seen as &nbsp;\\( \\var{b4}x^0 \\) so will differentiate to \\( 0 \\times \\var{b4} x^{0-1} \\) which, of course equals zero.&nbsp;</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } =\\simplify{&nbsp; {m1}*{b1}*x^({m1}-1) + {m2}*{b2}*x^({m2}-1)+ {m3}*{b3}*x^({m3}-1)&nbsp; } \\)</p>\n<p style=\"padding-left: 40px;\"></p>\n<p style=\"text-align: left;\">c)</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{y= {c1} x^{{p1}} + {c2} x^{{p2}} } \\)</p>\n<p style=\"padding-left: 40px;\">rem \\( \\frac{dy}{dx } = \\var{p1} \\times \\var{c1} x^{\\var{p1} - 1}&nbsp; +\\var[fractionNumbers]{p2} \\times \\var{c2} x^{\\var[fractionNumbers]{p2} - 1}&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">\\( \\frac{dy}{dx } = \\simplify{{p1}*{c1}*x^{{p1}-1} + {p2}*{c2}*x^{{p2}-1} } \\)</p>\n<p style=\"padding-left: 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