{"id":4258,"date":"2013-12-10T12:05:06","date_gmt":"2013-12-10T03:05:06","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4258"},"modified":"2014-08-01T19:26:06","modified_gmt":"2014-08-01T10:26:06","slug":"derivatives","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4258","title":{"rendered":"Derivatives"},"content":{"rendered":"<div class=\"theContentWrap-ccc\"><blockquote>\n<h3>Derivatives<\/h3>\n<p>The derivative of y = f(x) at a point x is defined as <\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+f%27%28x%29+%3D+%5Clim%5Climits_%7Bh+%5Crightarrow+0%7D%5Cfrac%7Bf%28x%2Bh%29+-+f%28x%29%7D%7Bh%7D+%3D+%5Clim%5Climits_%7B%5CDelta+x+%5Crightarrow+0%7D%5Cfrac%7B%5CDelta+y%7D%7B%5CDelta+x%7D+%3D+%5Cfrac%7Bdy%7D%7Bdx%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle f&#039;(x) = \\lim\\limits_{h \\rightarrow 0}\\frac{f(x+h) - f(x)}{h} = \\lim\\limits_{\\Delta x \\rightarrow 0}\\frac{\\Delta y}{\\Delta x} = \\frac{dy}{dx}' title='\\displaystyle f&#039;(x) = \\lim\\limits_{h \\rightarrow 0}\\frac{f(x+h) - f(x)}{h} = \\lim\\limits_{\\Delta x \\rightarrow 0}\\frac{\\Delta y}{\\Delta x} = \\frac{dy}{dx}' class='latex' \/><\/p>\n<p>where h = &Delta;x, &Delta;y = f(x + h) &#8211; f(x) = f(x + &Delta;x) &#8211; f(x) provided the limit exists. <\/p>\n<h3>Differentiation formulas<\/h3>\n<p>In the following u, v represent function of x while a, c, p represent constants. It&#8217;s assumed that the derivatives of u and v exist, i.e. u and v are differentiable. <\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7Bd%7D%7Bdx%7D%28u+%5Cpm+v%29+%3D+%5Cfrac%7Bdu%7D%7Bdx%7D+%5Cpm+%5Cfrac%7Bdv%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%28cu%29+%3D+c%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%28uv%29+%3D+u%5Cfrac%7Bdv%7D%7Bdx%7D+%2B+v%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Cleft%28%5Cfrac%7Bu%7D%7Bv%7D%5Cright%29+%3D+%5Cfrac%7Bv%28du%2Fdx%29+-+u%28dv%2Fdx%29%7D%7Bv%5E2%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7Du%5Ep+%3D+pu%5E%7Bp-1%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%28a%5Eu%29+%3D+a%5Eu%5Cln%7Ba%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7De%5Eu+%3D+e%5Eu%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Cln%7Bu%7D+%3D+%5Cfrac%7B1%7D%7Bu%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Csin%7Bu%7D+%3D+%5Ccos%7Bu%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Ccos%7Bu%7D+%3D+-%5Csin%7Bu%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Ctan%7Bu%7D+%3D+%5Csec%5E2%7Bu%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Csin%5E%7B-1%7Du+%3D+%5Cfrac%7B1%7D%7B%5Csqrt%7B1+-+u%5E2%7D%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Ccos%5E%7B-1%7Du+%3D+%5Cfrac%7B-1%7D%7B%5Csqrt%7B1+-+u%5E2%7D%7D%5Cfrac%7Bdu%7D%7Bdx%7D%5C%5C%5Cvspace%7B0.2+in%7D+++%5Cfrac%7Bd%7D%7Bdx%7D%5Ctan%5E%7B-1%7Du+%3D+%5Cfrac%7B1%7D%7B%5Csqrt%7B1+%2B+u%5E2%7D%7D%5Cfrac%7Bdu%7D%7Bdx%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle \\frac{d}{dx}(u \\pm v) = \\frac{du}{dx} \\pm \\frac{dv}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}(cu) = c\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}(uv) = u\\frac{dv}{dx} + v\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\left(\\frac{u}{v}\\right) = \\frac{v(du\/dx) - u(dv\/dx)}{v^2}\\\\\\vspace{0.2 in}   \\frac{d}{dx}u^p = pu^{p-1}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}(a^u) = a^u\\ln{a}\\\\\\vspace{0.2 in}   \\frac{d}{dx}e^u = e^u\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\ln{u} = \\frac{1}{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\sin{u} = \\cos{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\cos{u} = -\\sin{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\tan{u} = \\sec^2{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\sin^{-1}u = \\frac{1}{\\sqrt{1 - u^2}}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\cos^{-1}u = \\frac{-1}{\\sqrt{1 - u^2}}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\tan^{-1}u = \\frac{1}{\\sqrt{1 + u^2}}\\frac{du}{dx}' title='\\displaystyle \\frac{d}{dx}(u \\pm v) = \\frac{du}{dx} \\pm \\frac{dv}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}(cu) = c\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}(uv) = u\\frac{dv}{dx} + v\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\left(\\frac{u}{v}\\right) = \\frac{v(du\/dx) - u(dv\/dx)}{v^2}\\\\\\vspace{0.2 in}   \\frac{d}{dx}u^p = pu^{p-1}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}(a^u) = a^u\\ln{a}\\\\\\vspace{0.2 in}   \\frac{d}{dx}e^u = e^u\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\ln{u} = \\frac{1}{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\sin{u} = \\cos{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\cos{u} = -\\sin{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\tan{u} = \\sec^2{u}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\sin^{-1}u = \\frac{1}{\\sqrt{1 - u^2}}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\cos^{-1}u = \\frac{-1}{\\sqrt{1 - u^2}}\\frac{du}{dx}\\\\\\vspace{0.2 in}   \\frac{d}{dx}\\tan^{-1}u = \\frac{1}{\\sqrt{1 + u^2}}\\frac{du}{dx}' class='latex' \/><\/p>\n<p>In the special case where u = x, the above formulas are simplified since in such case du\/dx = 1. <\/p>\n<\/blockquote>\n<p><iframe src=\"\/\/rcm-fe.amazon-adsystem.com\/e\/cm?lt1=_blank&#038;bc1=000000&#038;IS2=1&#038;bg1=FFFFFF&#038;fc1=000000&#038;lc1=0000FF&#038;t=fujiitoshiki-22&#038;o=9&#038;p=8&#038;l=as4&#038;m=amazon&#038;f=ifr&#038;ref=ss_til&#038;asins=0071635408\" style=\"width:120px;height:240px;\" scrolling=\"no\" marginwidth=\"0\" marginheight=\"0\" frameborder=\"0\"><\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>Derivatives The derivative of y = f(x) at a point x is defined as where h = &Delta;x, &Delta;y = f(x + h) &#038;#82 &hellip; <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4258\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;Derivatives&#8221; \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-4258","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4258","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4258"}],"version-history":[{"count":8,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4258\/revisions"}],"predecessor-version":[{"id":6083,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4258\/revisions\/6083"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4258"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4258"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}