怎么求n阶函数 求函数的n阶导数
\u8fd9\u4e2a\u7684n\u9636\u5bfc\u6570\u5e94\u8be5\u600e\u4e48\u6c42\u5bf9\u4e8e\u51fd\u6570\u4e58\u79efy=f(x)*g(x)\u7684n\u9636\u5bfc\u6570\u6709\u5c55\u5f00\u516c\u5f0f\uff1a
y(n)=c(n,0)f(x)g(x)(n)+c(n,1)f(x)(1)g(x)(n-1)+c(n,2)f(x)(2)g(x)(n-2)+.c(n,n)f(x)(n)g(x).
\u5176\u4e2d:
y(n)\u8868\u793ay\u7684n\u9636\u5bfc\u6570,c(n,0)\u662f\u6392\u5217\u7ec4\u5408,f(x)(n)\u8868\u793af(x)\u7684n\u9636\u5bfc\u6570,g(x)(n)\u8868\u793ag(x)\u7684n\u9636\u5bfc\u6570.
\u5bf9\u4e8e\u672c\u9898\uff1a
f(x)=x^2,g(x)=sin2x
f(x)(1)=2x,f(x)(2)=2,f(x)(3)=0
\u6240\u4ee5\uff1a
y(50)=c(50,0)*x^2*(sin2x)(50)+c(50,1)*(2x)*(sin2x)(49)+c(50,2)*2*(sin2x)(48).
=x^2(sin2x)(50)+100x*(sin2x)(49)+2450(sin2x)(48).
\u4e0a\u9762\u8fd8\u53ef\u4ee5\u7ee7\u7eed\u5316\u7b80\uff0c\u6b63\u5f26\u51fd\u6570\u7684n\u9636\u5bfc\u6570\u662f\u53ef\u4ee5\u5c55\u5f00\u7684\uff0c\u6709\u516c\u5f0f\uff0c\u6211\u60f3\u4f60\u540e\u9762\u77e5\u9053\u7684\uff0c\u518d\u7b97\u51e0\u6b65\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u7ec8\u7ed3\u679c\u7684\uff0c\u5c31\u662f2^50[-x²sin2x+50xcos2x+(1225/2)sin2x]
\u5982\u56fe
y的n阶导数=[(-1)(-2)(-3).(-n) × (x-2)^(-n-1)] - [(-1)(-2)(-3).(-n) × (x-1)^(-n-1)]
=[(-1)^n]×n!×[ (x-2)^(-n-1) - (x-1)^(-n-1)]
a^b 就是a的b次方
n!就是n的阶乘=1×2×3×4.×(n-1)×n
y = 1/(x²-3x+2) = 1/[(x-2)(x-1)] = 1/(x-2) - 1/(x-1)
1/(x-2) 的 1 阶导数为 (-1)*(x-2)^(-2)
1/(x-2) 的 2 阶导数为 (-1)*(-2)*(x-2)^(-3)
.........................................
1/(x-2) 的 n 阶导数为 (-1)*(-2)*...*(-n)*(x-2)^(-(n+1))
即 1/(x-2) 的 n 阶导数为 (-1)^n * n! * (x-2)^[-(n+1)]
同理 1/(x-1) 的 n 阶导数为 (-1)^n * n! * (x-1)^[-(n+1)]
所以 f(x) 的 n 阶导数为 ：
(-1)^n * n! * (x-2)^[-(n+1)] - (-1)^n * n! * (x-1)^[-(n+1)]
即 (-1)^n * n! * {(x-2)^[-(n+1)] - (x-1)^[-(n+1)]}
如图所示
绛旓細姹俷闃瀵煎悗鎴愪负(n+1)!x 绗簩楂樻鏁伴」涓-(1+2+3+鈥︹+n)x^n 姹俷闃跺鍚庡彇绯绘暟鎴愪负-n(n+1)/2 鎵浠鐨刵闃跺鏁颁负(n+1)!x-n(n+1)/2 y=x(x-1)(x-2)(x-3)鈥︹(x-n)n闃跺鏁颁负(n+1)!x-n(n+1)/2 姹傚鐨勬剰涔夛細涓涓鍑芥暟鍦ㄦ煇涓鐐圭殑瀵兼暟鎻忚堪浜嗚繖涓嚱鏁板湪杩欎竴鐐归檮杩戠殑...
绛旓細鍑芥暟鐨n闃瀵兼暟鎬庝箞姹锛屽涓嬶細浜岄樁瀵兼暟鏄鏁扮殑瀵兼暟,灏嗗鏁板啀姹備竴娆″銆備笁闃跺氨鏄鏁扮殑瀵兼暟鐨勫鏁,姹傚涓夋銆俷闃跺鏁板氨鏄姹俷娆″銆傜畝鍗曠殑瑙勫緥鏈夛細x^n鐨刴闃跺鏁版槸n(n-1)鈥︹(n-m+1)x^(n-m)銆乪^x鐨刵闃跺鏁颁粛鏄痚^x銆乻inx鐨刵闃跺鏁版槸sin(x-n蟺/2蟺)銆乧osx鐨刵闃跺鏁版槸cos(x-n蟺/...
绛旓細鍑芥暟鍙鐨勬潯浠讹細濡傛灉涓涓嚱鏁扮殑瀹氫箟鍩熶负鍏ㄤ綋瀹炴暟锛屽嵆鍑芥暟鍦ㄥ叾涓婇兘鏈夊畾涔夈傚嚱鏁板湪瀹氫箟鍩熶腑涓鐐瑰彲瀵奸渶瑕佷竴瀹氱殑鏉′欢锛氬嚱鏁板湪璇ョ偣鐨勫乏鍙冲鏁板瓨鍦ㄤ笖鐩哥瓑锛屼笉鑳借瘉鏄庤繖鐐瑰鏁板瓨鍦ㄣ傚彧鏈夊乏鍙冲鏁板瓨鍦ㄤ笖鐩哥瓑锛屽苟涓斿湪璇ョ偣杩炵画锛屾墠鑳借瘉鏄庤鐐广n闃瀵兼暟鐨勬剰涔 浠庢蹇典笂璁诧紝楂橀樁瀵兼暟璁＄畻灏辨槸杩炵画杩涜涓闃跺鏁扮殑璁＄畻銆
绛旓細y=cos^2x y锛0锛=y=cos^2x y'=2cosx*(-sinx)=-2sinxcosx=-sin2x y''=-cos2x*2=-2cos2x y'''=-2x(-sin2x)x2=-4sin2x y(4)=-4xcos2x*2=-8cos2x y^(5)=-8(-sin2x)x2=-16sin2x y(6)=-16cos2x*2=-32cos2x n涓鸿杩帮紝y(n)=-2^(n-1)sin2x,n=2k-1,k:N ...
绛旓細1/(x-i)杩欑绠鍗鍑芥暟鐨n闃瀵兼暟锛岃繖閲岀殑i濮嬬粓褰撳父鏁扮湅寰呭嵆鍙傜敤澶嶆暟姹傚嚭鐨刵闃跺鏁版槸 (-1)^n * 2i * (1/(x-i)^锛坣+1锛- 1/(x+i)^(n+1))濡傛灉闈炶鎯虫秷鎺塱鐨勮瘽锛屾嫭鍙烽噷杈归氬垎锛屽垎姣嶆槸 锛坸^2+1锛塣(n+1)鍒嗗瓙鏄 锛坸+i锛塣(n+1) - (x-i)^(n+1)鍒╃敤 a^(n+1)-b^...
绛旓細鑾卞竷灏艰尐姹傚娉曞垯n闃鍏紡锛氳鍑芥暟u(x)銆乿(x)鍦ㄧ偣x閮藉叿鏈 n 闃瀵兼暟銆備簩闃跺鏁颁箻绉殑杩愮畻娉曞垯鏈夛細[u(x)*v(x)]''=u''(x)v(x)+2u'(x)v'(x)+u(x)v''(x)锛屽彲瑙佸鏁伴樁鏁拌秺楂橈紝鐩稿簲涔樼Н鐨勫鏁拌秺澶嶆潅锛屼絾鍏堕棿鍗存湁鐫鏄庢樉鐨勮寰嬫э紝涓哄綊绾冲叾涓鑸寰嬶紝涔樼Н鐨 n 闃跺鏁扮殑绯绘暟鍙婂鏁伴樁鏁...
绛旓細濡傛灉鍑芥暟鑳借〃绀轰负涓や釜绠鍗曞嚱鏁扮殑绉椂锛屽彲浠ユ牴鎹n闃姹傚鐨勮幈甯冨凹鍏瑰叕寮.鏈変竴浜涘彲浠ユ牴鎹墠鍑犻樁瀵兼暟鐢卞綊绾虫硶鎺ㄥ嚭瀹冪殑楂橀樁瀵兼暟 y ' = 2sinxcosx = sin2x y '' = 2cos2x y ''' = -4sin2x y^(4) = -8cos2x 涓鑸湴锛寉^(n) = 2^(n-1) * sin[2x+(n-1)鍏/2]...
绛旓細y ' = 2sinxcosx = sin2x y '' = 2cos2x y ''' = -4sin2x y^(4) = -8cos2x 涓鑸湴锛寉^(n) = 2^(n-1) * sin[2x+(n-1)鍏/2]渚嬪锛歗^^y=lnx/x y'=(1-lnx)/x^2=1/x^2-lnx/x^2 y"=-2/x^3-(1-2lnx)/x^3=-3/x^3+2lnx/x^3 璁皔(n)=(-1)^(n...
绛旓細澶氶」寮鍑芥暟鐨n闃瀵兼暟鐨勫叧閿槸澶т簬绛変簬n娆℃柟鐨勯」銆傚垎寮忓嚱鏁帮紝鍏蜂綋鎯呭喌锛屽叿浣撳垎鏋愩傝鎯呭鍥炬墍绀:渚涘弬鑰冿紝璇风瑧绾炽
绛旓細涓や釜鍑芥暟鐩镐箻鐨n闃姹傚锛屽彲浠ヤ娇鐢ㄨ幈甯冨凹鑼ㄥ叕寮忥紝璁＄畻杩囩▼濡備笅 1銆亁^2鍜宑os2x鐨刵闃跺鏁板涓嬶細2銆佷唬鍏ユ帹瀵笺
