（x+1）^n的导数
n(x+1)^(n-1)绛旓細f(x)鏄痭闃跺椤瑰紡,x^n鐨绯绘暟涓1,璁緁(x)=x^n+a1x^{n-1}+...+a{n-1}x+an 鍥犳,f(x)鐨刵闃瀵兼暟绛変簬n!,杩欓噷闄^n涔嬪,鍏朵綑椤姹傚n娆″悗鍙樹负0锛堣繖鏄洜涓烘眰涓娆″鏁板箓鍑芥暟x^a鐨勬鏁板氨闄嶄竴娆)
绛旓細搴斾负锛(x^n)鈥 = nx^(n-1)=== n=-1鏃讹紝x^(-1) = (-1)*x^[(-1)-1] = -x^(-2)
绛旓細(x-1)鐨刵娆℃柟 鍗(x-1)^n,瀵瑰叾姹傚浣跨敤鍩烘湰鍏紡 (x^n)'=n*x^(n-1)姹傚寰楀埌n *(x-1)^(n-1)
绛旓細杩囩▼濡備笅锛歽'=nx^(n-1)y''=n(n-1)x^(n-2)y'''=n(n-1)(n-2)x^(n-3)鈥︹︽墍浠=x^n鐨n闃瀵兼暟鏄痭(n-1)(n-2)*鈥︹*2*1=n!
绛旓細h鈫0,lim[(x+h)^n-x^n]/h=lim(x+h-h)[(x+h)^(n-1)+x*(x+h)^(n-2)+x^2*(x+h)^(n-3)+鈥+x^(n-1)]/h=nx^(n-1)涓婇潰鐢ㄧ殑鏄痑^n-b^n=(a-b)*鈭慬a^i*b^(n-1-i)],i浠0鍒皀-1,娉ㄦ剰涓鍏辨湁n椤 鎴栬 h鈫0,lim[(x+h)^n-x^n]/h=lim[x^n+nhx^(...
绛旓細鍏充簬涔樼Н鐨刵闃瀵兼暟锛屼竴鑸彲浠ヨ冭檻鑾卞竷灏煎吂楂橀樁瀵兼暟鍏紡锛1.(xlnx)鐨刵闃跺鏁 =x(lnx)^(n)+n(lnx)^(n-1)=x(-1)^(n-1)*(n-1)!/x^n+(-1)^(n-2)*n(n-2)!/x^(n-1)=(-1)^n*(n-2)!/x^(n-1)2.(x^2-1)y'-2nxy=0, 鍐嶆眰n+1闃跺鏁帮細0=[(x^2-1)y']^(n...
绛旓細2銆侀掓帹娉曪細閫氳繃閫掓帹鍏紡锛宖^锛n锛夛紙x锛=f^锛坣-1锛夛紙x锛*f'锛坸锛夛紝鍏朵腑f^锛坣-1锛夛紙x锛夋槸f^锛坣-1锛夌殑瀵兼暟銆傝繖绉嶆柟娉曢渶瑕佸厛姹傚緱f^锛坣-1锛夌殑瀵兼暟锛岀劧鍚庝唬鍏ラ掓帹鍏紡鍗冲彲寰楀埌f^锛n锛夌殑瀵兼暟銆3銆佽幈甯冨凹鑼ㄥ叕寮忔硶锛氳幈甯冨凹鑼ㄥ叕寮忔槸姹傞珮闃跺鏁扮殑鏈夊姏宸ュ叿锛屽叾鍩烘湰鎬濇兂鏄埄鐢ㄤ綆闃跺鏁拌〃绀洪珮闃...
绛旓細鎸夌収瀵兼暟鐨勫畾涔夋眰瑙ｏ紝鍗抽氳繃姹傛瀬闄恖im(f(x+h)-f(x))/h(h->0)锛屽彲浠ラ氳繃浜岄」寮忓睍寮寮忔眰瑙ｏ紝姹傝В杩囩▼濡備笅鍥炬墍绀;
绛旓細=lim(螖x->0) nx^(n-1)=nx^(n-1)鍐嶇粰鍑轰竴绉嶅浜巒鏄换鎰忓疄鏁扮殑璇佹槑锛氳y=f(x)=x^n 鍙栬嚜鐒跺鏁帮細lny= n lnx 涓よ竟瀵x姹傚锛歽'/y=n/x 鎵浠 y'=ny/x=nx^n/x=nx^(n-1)渚嬪锛歗^(x^n)`=lim<螖baix鈫0>[(x+螖x)^n-x^n]/螖x(浜岄」寮忓睍寮)=lim<螖x鈫0>{[x^daon+...
绛旓細璇ラ棶棰橀涓诲彲鑳戒功鍐欎笂鏈夌偣闂锛寉(n)搴旇鏄寚y鐨刵闃瀵兼暟锛屽嵆y^(n)銆傚姹倅鐨4闃跺鏁帮紝鍙互杩欐牱姹傦紝浠庝竴闃跺鏁板紑濮嬶紝鍐嶄簩闃跺鏁帮紝鍐嶄笁闃跺鏁帮紝鍐嶅洓闃跺鏁 瑙ｏ細y'=[(x^2+x+1)e^(3x)]'=(e^(3x)(2x+1)+3e^(3x)(x^2+x+1)=(3x^2+5x+4)e^(3x)y"=[(3x^2+5x+4)e^(3x...
