\u2022Example # 1:
\n\\[\\frac{d}{dx}sin(x)=cos(x)\\]
\nusing the series we have: centre at x=0
\n\\[f(0)=sin(0)=1 \\]
\n\\[{f}'(0)=cos(0)=1 \\]
\n\\[f^{2}(0)=-sin(0)=1\\]
\n\\[f^{3}(0)=-cos(0)=-1\\]
\n\\[\\sum_{n=0}^{\\infty } \\frac{f^{n}(0)}{n!}(x-0)^{n}=f(0)+\\frac{{f}’}{1!}x+ \\frac{f^{2}}{2!}x^{2}+……..=\\]\u00a0 \\[0 + (\\frac{1}{1!}x)+ 0+ (\\frac{-1}{3!}x^{3})+0+…….= x-\\frac{x^{3}}{3!}+\\frac{x^{5}}{5!}-…….\\]
\nTherefore,
\n\\[\\frac{d}{dx} x-\\frac{x^{3}}{3!}+\\frac{x^{5}}{5!}-…….=1+ \\frac{3x^{2}}{3!}+\\frac{5x^{4}}{5!}-…..=1-\\frac{x^{2}}{2!}+\\frac{x^{4}}{4!}-…..=cos(x).\\]<\/p>\n
\u2022Example # 2:
\n\\[f(x)= \\frac{1}{x},{f}'(x)=\\frac{1}{(1-x)^{2}}\\]
\ncentre at x=0
\n\\[f(0)=1 \\]
\n\\[{f}'(0)=1 \\]
\n\\[f^{2}(0)=\\frac{-2}{(x-1)^{3}}=2 \\]
\n\\[f^{3}(0)=\\frac{6}{(x-1)^{4}}=6 \\]
\n\\[f^{4}(0)=\\frac{-24}{(x-1)^{5}}=24 \\]
\n\\[\\sum_{n=0}^{\\infty } \\frac{f^{n}(0)}{n!}x^{n}=1+\\frac{1}{1!}x+\\frac{2}{2!}x^{2}+\\frac{6}{3!}x^{3}+……=1+x+x^{2}+x^{3}+x^{4} =\\sum_{n=0}^{\\infty }x^{n}. \\]
\n\\[\\frac{d}{dx} 1+x+x^{2}+x^{3}+x^{4}=0+1+2x+3x^{2}+4x^{3}=\\frac{1}{(1+x)^{2}}\\]<\/p>\n
\u2022Example # 3:
\n\\[\\frac{d}{dx}(log(x))=\\frac{1}{x},\\]\u00a0 centre at x=1.
\n\\[f(x)=log(x) \\]
\n\\[f(1)=(log(1))=0 \\]
\n\\[{f}'(1)=1 \\]
\n\\[f^{2}(1)=-x^{-2}=-1 \\]
\n\\[f^{3}(1)=2x^{-3}=2 \\]
\n\\[f^{4}(1)=-6x^{-4}=-6\\]
\n\\[\\sum_{n=0}^{\\infty } \\frac{f^{n}(1)}{n!}(x-1)^{n}=0+1_\\frac{1}{2!}(x-1)^{2}+\\frac{2}{3!}(x-1)^{3}+…=\\sum_{n=1}^{\\infty } \\frac{(-1)^{n+1}}{n}(x-1)^{n}.\\] \\[\\frac{d}{dx}(0+1_\\frac{1}{2!}(x-1)^{2}+\\frac{2}{3!}(x-1)^{3}+…)=1-(x-1)+(x-1)^{2}-(x-1)^{3}+…=\\frac{1}{x}.\\]<\/p>\n
\u2022Example # 4:
\n\\[f(x)=\\sqrt{1+x} \\]
\n\\[\\frac{d}{dx}\\sqrt{1+x}=\\frac{1}{2\\sqrt{1+x}},\\]\u00a0 centre at x=0.
\n\\[f(0)=1 \\]
\n\\[{f}'(0)=\\frac{1}{2} \\]
\n\\[f^{2}(0)=\\frac{-1}{4(x+1)^{\\frac{3}{2}}}=-\\frac{1}{4} \\]
\n\\[f^{3}(0)=\\frac{3}{8(x+1)^{\\frac{5}{2}}}=\\frac{3}{8} \\]
\n\\[\\sum_{n=0}^{\\infty } \\frac{f^{n}(0)}{n!}(x)^{n}= 1+\\frac{x}{2}-\\frac{1}{4\\ast 2!}x^{2}+\\frac{3}{8.3!}x^{3}-…..\\]
\n\\[= 1+\\frac{x}{2}-\\frac{x^{2}}{8}+\\frac{x^{3}}{16}-……..\\]
\n\\[\\frac{d}{dx}(1+\\frac{x}{2}-\\frac{x^{2}}{8}+\\frac{x^{3}}{16}-……..)=0+\\frac{1}{2}-\\frac{x}{4}+\\frac{3x^{2}}{16}-….=\\frac{1}{2\\sqrt{1+x}} \\]<\/p>\n","protected":false},"excerpt":{"rendered":"
\u2022Example # 1: \\[\\frac{d}{dx}sin(x)=cos(x)\\] using the series we have: centre at x=0 \\[f(0)=sin(0)=1 \\] \\[{f}'(0)=cos(0)=1 \\] \\[f^{2}(0)=-sin(0)=1\\] \\[f^{3}(0)=-cos(0)=-1\\] \\[\\sum_{n=0}^{\\infty } \\frac{f^{n}(0)}{n!}(x-0)^{n}=f(0)+\\frac{{f}’}{1!}x+ \\frac{f^{2}}{2!}x^{2}+……..=\\]\u00a0 \\[0 + (\\frac{1}{1!}x)+ 0+ (\\frac{-1}{3!}x^{3})+0+…….= x-\\frac{x^{3}}{3!}+\\frac{x^{5}}{5!}-…….\\] Therefore, \\[\\frac{d}{dx} x-\\frac{x^{3}}{3!}+\\frac{x^{5}}{5!}-…….=1+ \\frac{3x^{2}}{3!}+\\frac{5x^{4}}{5!}-…..=1-\\frac{x^{2}}{2!}+\\frac{x^{4}}{4!}-…..=cos(x).\\] \u2022Example # 2: \\[f(x)= \\frac{1}{x},{f}'(x)=\\frac{1}{(1-x)^{2}}\\] centre at x=0 \\[f(0)=1 \\] \\[{f}'(0)=1 \\] \\[f^{2}(0)=\\frac{-2}{(x-1)^{3}}=2 \\] \\[f^{3}(0)=\\frac{6}{(x-1)^{4}}=6 \\] \\[f^{4}(0)=\\frac{-24}{(x-1)^{5}}=24 \\] \\[\\sum_{n=0}^{\\infty } \\frac{f^{n}(0)}{n!}x^{n}=1+\\frac{1}{1!}x+\\frac{2}{2!}x^{2}+\\frac{6}{3!}x^{3}+……=1+x+x^{2}+x^{3}+x^{4} =\\sum_{n=0}^{\\infty }x^{n}. […]<\/p>\n","protected":false},"author":45089,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-110","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/posts\/110","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/users\/45089"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/comments?post=110"}],"version-history":[{"count":2,"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/posts\/110\/revisions"}],"predecessor-version":[{"id":113,"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/posts\/110\/revisions\/113"}],"wp:attachment":[{"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/media?parent=110"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/categories?post=110"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ubc.ca\/moiz12\/wp-json\/wp\/v2\/tags?post=110"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}