arcsinx的导数 y=arcsinx怎么求导啊，麻烦详细点
\u8bf7\u6559\u5982\u4f55\u6c42arcsinX\u7684\u5bfc\u6570\uff1fy=arcsinx y'=1/\u221a\uff081-x^2\uff09
\u53cd\u51fd\u6570\u7684\u5bfc\u6570\uff1a
y=arcsinx
\u90a3\u4e48\uff0csiny=x
\u6c42\u5bfc\u5f97\u5230\uff0ccosy *y'=1
\u5373 y'=1/cosy=1/\u221a[1-(siny)^2]=1/\u221a(1-x^2)
(arcsinx)'=1/(1-x^2)^1/2
(arccosx)'=-1/(1-x^2)^1/2
(arctanx)'=1/(1+x^2)
(arccotx)'=-1/(1+x^2)
(arcsecx)'=1/(|x|(x^2-1)^1/2)
(arccscx)'=-1/(|x|(x^2-1)^1/2)
\u4e0d\u662f\u6240\u6709\u7684\u51fd\u6570\u90fd\u6709\u5bfc\u6570\uff0c\u4e00\u4e2a\u51fd\u6570\u4e5f\u4e0d\u4e00\u5b9a\u5728\u6240\u6709\u7684\u70b9\u4e0a\u90fd\u6709\u5bfc\u6570\u3002\u82e5\u67d0\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u5bfc\u6570\u5b58\u5728\uff0c\u5219\u79f0\u5176\u5728\u8fd9\u4e00\u70b9\u53ef\u5bfc\uff0c\u5426\u5219\u79f0\u4e3a\u4e0d\u53ef\u5bfc\u3002\u7136\u800c\uff0c\u53ef\u5bfc\u7684\u51fd\u6570\u4e00\u5b9a\u8fde\u7eed\uff1b\u4e0d\u8fde\u7eed\u7684\u51fd\u6570\u4e00\u5b9a\u4e0d\u53ef\u5bfc\u3002
\u6269\u5c55\u8d44\u6599
\u5546\u7684\u5bfc\u6570\u516c\u5f0f\uff1a
(u/v)'=[u*v^(-1)]'
=u' * [v^(-1)] +[v^(-1)]' * u
= u' * [v^(-1)] + (-1)v^(-2)*v' * u
=u'/v - u*v'/(v^2)
\u901a\u5206\uff0c\u6613\u5f97\uff1a
(u/v)=(u'v-uv')/v²
\u5e38\u7528\u5bfc\u6570\u516c\u5f0f\uff1a
1\u3001y=c(c\u4e3a\u5e38\u6570) y'=0
2\u3001y=x^n y'=nx^(n-1)
3\u3001y=a^x y'=a^xlna\uff0cy=e^x y'=e^x
4\u3001y=logax y'=logae/x\uff0cy=lnx y'=1/x
5\u3001y=sinx y'=cosx
6\u3001y=cosx y'=-sinx
7\u3001y=tanx y'=1/cos^2x
8\u3001y=cotx y'=-1/sin^2x
9\u3001y=arcsinx y'=1/\u221a1-x^2
\u4f7f\u7528\u53cd\u51fd\u6570\u53ef\u4ee5\u5bf9y=arcsinx\u6c42\u5bfc\uff1a
\u56e0\u4e3ay=arcsinx\uff0c\u6240\u4ee5\u5f97\u5230
siny=x \u7b49\u5f0f\u4e24\u8fb9\u5bf9x\u6c42\u5bfc
y'cosy=1
\u53ef\u5f97y'=1/cosy=1/\u221a(1-sin^2(y))
\u53ef\u5f97y'= 1/\u221a(1-x^2)
\u4e09\u89d2\u51fd\u6570\u7684\u6c42\u5bfc\u9700\u8981\u7528\u5230\u7684\u5f0f\u5b50\uff1a(sinx)'=cosx\u3001(cosx)'=-sinx\u3001(tanx)'=sec²x=1+tan²x\u3001(cotx)'=-csc²x\u3001(secx)' =tanx\u00b7secx\u3001(cscx)' =-cotx\u00b7cscx.\u3001(tanx)'=(sinx/cosx)'=sec²x\u3002
\u6269\u5c55\u8d44\u6599\u53c2\u6570\u8868\u8fbe\u5f0f\u6c42\u5bfc\u6cd5\u5219\uff1a
\u82e5\u53c2\u6570\u8868\u8fbe\uff0c\u4e3a\u4e00\u4e2ay\u5173\u4e8ex\u7684\u51fd\u6570\uff0c\u7531\u51fd\u6570\u89c4\u5f8b\u7684x\uff0c\u800c\u8fd9\u4e2ax\u503c\u7684\u90a3\u4e2at\u8981\u5bf9\u5e94\u552f\u4e00\u7684\u4e00\u4e2ay\u503c\uff0c\u624d\u80fdy\u4e3ax\u7684\u51fd\u6570\u3002\u7531\u6b64\u53ef\u89c1\u5fc5\u5b58\u5728\u53cd\u51fd\u6570\uff0c\u4e8e\u662f\u4ee3\u5165\u3002
\u82e5\u4e2d\u5b58\u5728\u9690\u51fd\u6570\uff0c\u8fd9\u91cc\u4ec5\u662f\u8bf4y\u4e3a\u4e00\u4e2ax\u7684\u51fd\u6570\u5e76\u975e\u8bf4y\u4e00\u5b9a\u88ab\u53cd\u89e3\u51fa\u6765\u4e3a\u663e\u5f0f\u8868\u8fbe\u3002\u5373\uff0c\u5c3d\u7ba1y\u672a\u53cd\u89e3\u51fa\u6765\uff0c\u53ea\u8981y\u5173\u4e8ex\u7684\u9690\u51fd\u6570\u5b58\u5728\u4e14\u53ef\u5bfc\uff0c\u6211\u4eec\u5229\u7528\u590d\u5408\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219\u5219\u4ecd\u53ef\u4ee5\u6c42\u51fa\u5176\u53cd\u51fd\u6570\u3002
\u53c2\u8003\u8d44\u6599\uff1a\u767e\u5ea6\u767e\u79d1\u2014\u6c42\u5bfc
(arcsinx)'=1/√(1-x^2)
如果不记得就用反函数的导数来推，
y=arcsinx，
那么
siny=x，
求导得到
cosy *y'=1
即
y'=1/cosy=1/√[1-(siny)^2]=1/√(1-x^2)
arc是神马东西……sinx的导数是cosx
绛旓細鏂规硶濡備笅锛岃浣滃弬鑰冿細鑻ユ湁甯姪锛岃閲囩撼銆
绛旓細arcsinx鐨勫鏁颁负1/鈭(1-x^2)銆傝В绛旇繃绋嬪涓嬶細姝や负闅愬嚱鏁版眰瀵硷紝浠=arcsinx 閫氳繃杞彉鍙緱锛歽=arcsinx锛岄偅涔坰iny=x銆備袱杈硅繘琛屾眰瀵硷細cosy 脳 y'=1銆傚嵆锛歽'=1/cosy=1/鈭歔1-(siny)^2]=1/鈭(1-x^2)銆傚悓鐞嗗彲寰楋細arccosx鐨勫鏁颁负-1/鈭(1-x^2)銆
绛旓細arcsinx鐨勫鏁版槸1/鈭氾紙1-x²锕氾紝鑰宎rccosx=蟺/2-arcsinx锛岄偅涔堝arccosx姹傚锛寉'=-1/鈭(1-x²)銆
绛旓細arcsinx鐨勫鏁(arcsinx)'=1/鏍瑰彿锛1-x^2锛銆傝y=arcsinx鈭圼-蟺/2,蟺/2]锛屽垯x=siny ,1=锛坈osy)*y' ,y'=1/cosy=1/鏍瑰彿锛1-sin^2y)=1/鏍瑰彿锛1-x^2锛夈俛rcsinx鐨勫鏁拌В绛旇繃绋嬶細1銆佸弽鍑芥暟鐨勫鏁颁笌鍘熷嚱鏁扮殑瀵兼暟鍏崇郴鏄鍘熷嚱鏁颁负y=fx锛屽垯鍏跺弽鍑芥暟鍦▂鐐圭殑瀵兼暟涓巉'x浜掍负鍊掓暟锛屽嵆鍘...
绛旓細arcsinx鐨勫鏁版槸y'=1/cosy=1/鈭歔1-(siny)²]=1/鈭(1-x²)鎺ㄥ杩囩▼璇存槑锛歽=arcsinx y'=1/鈭氾紙1-x²锛夊弽鍑芥暟鐨勫鏁帮細y=arcsinx,閭ｄ箞锛宻iny=x,姹傚寰楀埌锛宑osy*y'=1 鍗硑'=1/cosy=1/鈭歔1-(siny)²]=1/鈭(1-x²)鍙嶄笁瑙掑嚱鏁颁粙缁 鍙嶄笁瑙掑嚱鏁版槸姝ｅ鸡锛...
绛旓細arcsinx鐨勫鏁1/鈭(1-x^2)銆傝В绛旇繃绋嬪涓嬶細姝や负闅愬嚱鏁版眰瀵硷紝浠=arcsinx 閫氳繃杞彉鍙緱锛歽=arcsinx锛岄偅涔坰iny=x銆備袱杈硅繘琛屾眰瀵硷細cosy 脳 y'=1銆傚嵆锛y'=1/cosy=1/鈭[1-(siny)^2]=1/鈭(1-x^2)銆
绛旓細arcsinx鐨勫鏁1/鈭(1-x^2)銆傝В绛旇繃绋嬪涓嬶細姝や负闅愬嚱鏁版眰瀵硷紝浠=arcsinx 閫氳繃杞彉鍙緱锛歽=arcsinx锛岄偅涔坰iny=x銆備袱杈硅繘琛屾眰zhuan瀵硷細cosy 脳 y'=1銆傚嵆锛y'=1/cosy=1/鈭[1-(siny)^2]=1/鈭(1-x^2)
绛旓細arcsinx鐨勫鏁版槸锛y'=1/cosy=1/鈭[1-(siny)²]=1/鈭(1-x²)锛屾涓洪殣鍑芥暟姹傚銆傛帹瀵艰繃绋 y=arcsinx y'=1/鈭氾紙1-x²锛夊弽鍑芥暟鐨勫鏁帮細y=arcsinx,閭ｄ箞锛宻iny=x,姹傚寰楀埌锛宑osy*y'=1 鍗硑'=1/cosy=1/鈭歔1-(siny)²]=1/鈭(1-x²)闅愬嚱鏁板鏁扮殑姹傝В ...
绛旓細y=arcsinx(-1<x<1)鏄痻=siny鐨勫弽鍑芥暟锛寈=siny鍗曡皟鍙锛屼笖siny鐨勫鏁颁负cosy>0 dy/dx=1/cosy=1/鏍瑰彿涓1-x^2 鎵浠arcsinx鐨勫鏁涓1闄ゆ牴鍙蜂笅1-x^2 鍙嶄笁瑙掑嚱鏁颁腑鐨勫弽姝ｅ鸡銆傛剰鎬濅负锛歴in(a) = b; 绛変环浜 arcsin(b) = a;瀹氫箟鍩燂細銆-1锛1銆戯紱鍊煎煙锛氥-pi / 2锛宲i / 2銆;鍙嶄笁瑙...
绛旓細arcsinx鐨勫鏁版槸锛y'=1/cosy=1/鈭[1-(siny)]=1/鈭(1-x)锛屾涓洪殣鍑芥暟姹傚銆傛帹瀵艰繃绋嬶細y=arcsinx锛寉'=1/鈭氾紙1-x锛夛紝鍙嶅嚱鏁扮殑瀵兼暟锛歽=arcsinx锛岄偅涔堬紝siny=x锛屾眰瀵煎緱鍒癱osy*y'=1銆 鎵╁睍璧勬枡 arcsinx瀵兼暟鐨'姹傝В锛氭柟娉1锛氬厛鎶婇殣鍑芥暟杞寲鎴愭樉鍑芥暟锛屽啀鍒╃敤鏄惧嚱鏁版眰瀵肩殑鏂规硶姹傚锛涙柟...
