{"id":8295,"date":"2016-12-26t09:20:46","date_gmt":"2016-12-26t17:20:46","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=8295"},"modified":"2016-12-28t11:46:27","modified_gmt":"2016-12-28t19:46:27","slug":"meant-differentiation-mathematics","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/meant-differentiation-mathematics\/","title":{"rendered":"what is meant by differentiation in mathematics?"},"content":{"rendered":"<p>differentiation comes down to figuring out how one variable changes with respect to another variable. if this change is a constant (as we have in a line), this concept becomes very similar to the idea of a slope.\u00a0 but calculus is all about curves, and differentiation allows us to figure out rates of change when this change is itself changing.<\/p>\n<h2>an example of differentiation<\/h2>\n<p>the best way to look at differentiation is to look at a real-world example.\u00a0 let us take the old physics question that asks us to model a car starting at 30 meters per second, but slamming on the breaks.\u00a0 intuitively, we should know something about the velocity and acceleration.\u00a0 calculus and the derivative will allow us to model this mathematically, and figure out what\u2019s changing at any point.<\/p>\n<p>if you\u2019ve taken a physics class, you should be able to understand the following equation:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8296\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-12.14.39-pm.png\" alt=\"differentiation mathematics\" width=\"133\" height=\"41\" \/><\/p>\n<p>where x(t) is our function for a position at any time t, in seconds.\u00a0 our initial speed is 30 meters per second, and every second we\u2019re slowing down by 5 meters per second.\u00a0 i\u2019ve graphed the function below:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8297\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-12.14.42-pm.png\" alt=\"differentiation mathematics\" width=\"283\" height=\"278\" \/><\/p>\n<p>clearly, our position is not a linear function (it is quadratic).\u00a0 on our x-axis, we have time (in seconds), and our y-axis, we have position.\u00a0 how fast our position changes with respect to time (distance\/time) is a rate of change more commonly referred to as velocity (or speed).\u00a0 now, if this was a linear function, our velocity would simply be the slope of the graph.\u00a0 however, since it is quadratic, we need to take the <a href=\"\/\/www.catharsisit.com\/hs\/ap\/derivatives-ap-calculus-ab-bc-exams\/\">derivative<\/a>.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8298\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-12.14.47-pm.png\" alt=\"differentiation mathematics\" width=\"183\" height=\"64\" \/><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-medium wp-image-8299\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-12.14.50-pm-251x300.png\" alt=\"differentiation mathematics\" width=\"251\" height=\"300\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-12.14.50-pm-251x300.png 251w, \/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-12.14.50-pm.png 281w\" sizes=\"(max-width: 251px) 100vw, 251px\" \/><\/p>\n<p>as we can see, our velocity is slowly decreasing.\u00a0 at t = 0, our velocity is 30 meters per second, but eventually goes to 0 by the 3<sup>rd<\/sup> second.<\/p>\n<p>if we want to look at our rate of change of our rate of change, what we will see is how our velocity changes with respect to time as well.\u00a0 our change in velocity per second is more commonly referred to as acceleration.\u00a0 since our velocity time graph is linear, our derivative will be the same as our slope in this case.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-8300\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/screen-shot-2016-12-26-at-12.14.56-pm.png\" alt=\"differentiation mathematics\" width=\"157\" height=\"34\" \/><\/p>\n<p>our acceleration is -10 m\/s<sup>2<\/sup>.\u00a0 our we can say that we\u2019re slowing down 10 m\/s every second.<\/p>\n<p>&nbsp;<\/p>\n<p>the derivative is an important tool in determining how variables change with respect to each other.\u00a0 ap calculus ab primarily deals with two dimensions, but we can also use derivatives to compare multiple dimensions.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>calculus is all about curves and differentiation allows us to figure out rates of change when the change is itself changing. click here to learn more!<\/p>\n","protected":false},"author":224,"featured_media":8297,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241,248,251,253,252,254],"ppma_author":[24930],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>what is meant by differentiation in mathematics? - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"calculus is all about curves and differentiation allows us to figure out rates of change when the change is itself changing. click here to learn more!\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"\/\/www.catharsisit.com\/hs\/ap\/meant-differentiation-mathematics\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta 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