{"id":37404,"date":"2018-01-24T14:42:35","date_gmt":"2018-01-24T14:42:35","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=37404"},"modified":"2026-03-26T11:37:37","modified_gmt":"2026-03-26T16:37:37","slug":"derivative-properties-and-formulas","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/derivative-properties-and-formulas\/","title":{"rendered":"Derivative Properties and Formulas"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_R9PjnGpKqoA\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_R9PjnGpKqoA\" data-source-videoID=\"R9PjnGpKqoA\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Derivative Properties and Formulas Video\" height=\"1080\" width=\"1920\" class=\"size-full\" data-matomo-title = \"Derivative Properties and Formulas\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_R9PjnGpKqoA:hover {cursor:pointer;} img#videoThumbnailImage_R9PjnGpKqoA {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/07\/139-thumb-final.jpg\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_R9PjnGpKqoA\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_R9PjnGpKqoA\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Derivative Properties and Formulas\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_R9PjnGpKqoA\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_R9PjnGpKqoA\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_R9PjnGpKqoA\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction Gsq_Function() {\n  var x = document.getElementById(\"Gsq\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"Gsq_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"Gsq\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Constant_Rule\" class=\"smooth-scroll\">Constant Rule<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Power_Rule\" class=\"smooth-scroll\">Power Rule<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Sum_and_Difference_Rule\" class=\"smooth-scroll\">Sum and Difference Rule<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Review\" class=\"smooth-scroll\">Review<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>\u201cProperty\u201d is a mathematical term for an identity or a true statement which can simplify a problem. For example, one of the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/laws-of-exponents\/\">exponent properties<\/a> is that \\(a^{x}a^{y}=a^{x+y}\\).<\/p>\n<p>In this video, we are going to go over some derivative properties that will help simplify many of the problems you\u2019ll come across.<\/p>\n<h2><span id=\"Constant_Rule\" class=\"m-toc-anchor\"><\/span>Constant Rule<\/h2>\n<p>\nThe first derivative property we are going to cover is called the constant rule, which states that the derivative of any constant is zero. <\/p>\n<div class=\"examplesentence\">\\(\\frac{d}{dx}k=0,\\text{ }k\\text{ } \\epsilon \\text{ }R\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/08\/I09898-Derivative-Properties-1-e1661531058846.png\" alt=\"derivative of a constant is 0\" width=\"400\" height=\"345\" class=\"alignnone size-full wp-image-138898\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/08\/I09898-Derivative-Properties-1-e1661531058846.png 400w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2022\/08\/I09898-Derivative-Properties-1-e1661531058846-300x259.png 300w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/p>\n<p>This makes sense, because if we graph a constant function, such as \\(f(x)=3\\), we get a horizontal line. Because the line is horizontal, the slope at all points is zero. This is true no matter what point we pick, whether large or small, positive or negative.<\/p>\n<p>As long as we are just talking about a constant, the derivative is zero.<\/p>\n<div class=\"examplesentence\">\\(\\frac{d}{dx}3=0\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Power_Rule\" class=\"m-toc-anchor\"><\/span>Power Rule<\/h2>\n<p>\nThe next property to consider is that we can factor out constants from derivatives. For example, we already know that the derivative of \\(3x^{2}\\) is \\(6x\\), because the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/definition-of-the-derivative\/\">power rule<\/a> tells us to multiply by the starting exponent and then reduce the power by one. <\/p>\n<div class=\"examplesentence\">\\(\\frac{d}{dx}x^{n}=nx^{n-1}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnother way to think of this, though, is that we can pull that 3 out to the front of the derivative\u2026<\/p>\n<div class=\"examplesentence\">\\(\\frac{d}{dx}3x^{2}=3\\frac{d}{dx}x^{2}\\)<\/div>\n<p>\n&nbsp;<br \/>\n\u2026take the derivative of \\(x^{2}\\) by itself\u2026<\/p>\n<div class=\"examplesentence\">\\(3(2x)\\)<\/div>\n<p>\n&nbsp;<br \/>\n\u2026and simplify to get the same result.<\/p>\n<div class=\"examplesentence\">\\(6x\\)<\/div>\n<p>\n&nbsp;<br \/>\nOf course, in this example, it\u2019s faster to just multiply 3 by 2 and reduce the power without the intermediate factoring step. But with trickier derivatives, like \\(\\frac{d}{dx}\\frac{5x^{3}}{3}\\), this property can be helpful.<\/p>\n<p>To take this derivative, notice that we can factor out the 5 from the top and the 3 from the bottom as \\(\\frac{5}{3}\\). Then we can handle \\(x^{3}\\) by itself.<\/p>\n<div class=\"examplesentence\">\\(\\frac{5}{3}\\cdot \\frac{d}{dx}x^{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe derivative of \\(x^{3}\\) is \\(3x^{2}\\), so we will write that now.<\/p>\n<div class=\"examplesentence\">\\(\\frac{5}{3}(3x^{2})\\)<\/div>\n<p>\n&nbsp;<br \/>\nFrom here, the 3s cancel and we are left with \\(5x^{2}\\). If you see a derivative problem with a coefficient\u2014even a fractional one\u2014you can use this factoring property to simplify the problem.<\/p>\n<h2><span id=\"Sum_and_Difference_Rule\" class=\"m-toc-anchor\"><\/span>Sum and Difference Rule<\/h2>\n<p>\nThe third and final derivative property we are going to cover in this video is the sum and difference rule. You will likely come across many derivative problems that involve functions with multiple terms. For example, you may be asked to find the derivative of \\(f(x)=x^{2}+4x-5\\). There\u2019s nothing to fear with such problems, though, because these derivatives can be handled one term at a time.<\/p>\n<p>The derivative of the first term, \\(x^{2}\\), is \\(2x\\). The second term\u2019s derivative is 4, and the third term\u2019s derivative is zero because it is simply a constant. So \\(f'(x)\\) is the sum of each term\u2019s derivatives, which gives us:<\/p>\n<div class=\"examplesentence\">\\(f'(x)=2x+4\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe formal way of expressing the sum and difference rule is that the derivative of many terms is equal to the sum (or difference, if the terms are subtracted) of each term\u2019s derivative.<\/p>\n<div class=\"examplesentence\">\\(\\frac{d}{dx}(f_{1}(x)+f_{2}(x)+\u2026)\\) \\(=(\\frac{d}{dx}f_{1}(x))\\pm (\\frac{d}{dx}f_{2}(x))\\pm\\)<\/div>\n<hr>\n<h2><span id=\"Review\" class=\"m-toc-anchor\"><\/span>Review<\/h2>\n<p>\nLet\u2019s quickly review what we\u2019ve discussed in this video.<\/p>\n<p>First, the constant rule tells us that the derivative of any constant is always 0. Second, we learned that some derivatives can be made easier by factoring out a constant and handling the remaining function on its own. <\/p>\n<p>Finally, the sum and difference rule states that the derivative of a function with many terms is simply the sum or difference of the individual terms\u2019 derivatives.<\/p>\n<p>I hope this video was helpful. Thanks for watching, and happy studying!<\/p>\n<\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/trigonometry\/\">Return to Trigonometry Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Trigonometry Videos<\/p>\n","protected":false},"author":1,"featured_media":130078,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-37404","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-calculus-videos","7":"page_category-math-advertising-group","8":"page_category-video-pages-for-study-course-sidebar-ad","9":"page_type-video","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/37404","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=37404"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/37404\/revisions"}],"predecessor-version":[{"id":280277,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/37404\/revisions\/280277"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/130078"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=37404"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}