y＝arcsin（1-x∧2）求微分 y=arcsin根号下(1-x^2),求微分
\u6c42y=arcsin\u221a\uff081-x^2\uff09\u7684\u5fae\u5206\uff0c\u4e3a\u4ec0\u4e48x\u8981\u6709\u7edd\u5bf9\u503c\u554a\u56e0\u4e3a\uff1a
\u4e0a\u56fe\u4e2d\u6a59\u8272\u6807\u8bb0\u5904\u662f\u5f00\u65b9\uff0c\u221a[1-1+x²]=\u221ax²\uff0c\u5f53x\u5927\u4e8e0\u7684\u65f6\u5019\uff0c\u6240\u5f97\u7684\u5fae\u5206\u4e0d\u52a0\u7edd\u5bf9\u503c\u4e0d\u5f71\u54cd\u6700\u7ec8\u7ed3\u679c\uff0c\u5f53x\u5c0f\u4e8e0\u7684\u65f6\u5019\uff0c\u5982\u679c\u4e0d\u52a0\u7edd\u5bf9\u503c\uff0c\u5f97\u5230\u7684\u5fae\u5206\u5f71\u54cd\u6700\u7ec8\u7ed3\u679c\u3002\u6240\u4ee5\u5fc5\u987b\u52a0\u7edd\u5bf9\u503c\u3002
\u5fae\u5206\u5728\u6570\u5b66\u4e2d\u7684\u5b9a\u4e49\uff1a\u7531\u51fd\u6570B=f(A)\uff0c\u5f97\u5230A\u3001B\u4e24\u4e2a\u6570\u96c6\uff0c\u5728A\u4e2d\u5f53dx\u9760\u8fd1\u81ea\u5df1\u65f6\uff0c\u51fd\u6570\u5728dx\u5904\u7684\u6781\u9650\u53eb\u4f5c\u51fd\u6570\u5728dx\u5904\u7684\u5fae\u5206\uff0c\u5fae\u5206\u7684\u4e2d\u5fc3\u601d\u60f3\u662f\u65e0\u7a77\u5206\u5272\u3002\u5fae\u5206\u662f\u51fd\u6570\u6539\u53d8\u91cf\u7684\u7ebf\u6027\u4e3b\u8981\u90e8\u5206\u3002\u5fae\u79ef\u5206\u7684\u57fa\u672c\u6982\u5ff5\u4e4b\u4e00\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u8bbe\u51fd\u6570y = f(x)\u5728\u67d0\u533a\u95f4\u5185\u6709\u5b9a\u4e49\uff0cx0\u53cax0+\u25b3x\u5728\u8fd9\u533a\u95f4\u5185\uff0c\u82e5\u51fd\u6570\u7684\u589e\u91cf\u0394y = f(x0 + \u0394x) − f(x0)\u53ef\u8868\u793a\u4e3a\u0394y = A\u0394x + o(\u0394x)\uff0c\u5176\u4e2dA\u662f\u4e0d\u4f9d\u8d56\u4e8e\u25b3x\u7684\u5e38\u6570\uff0c o(\u0394x)\u662f\u25b3x\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f\uff0c\u5219\u79f0\u51fd\u6570y = f(x)\u5728\u70b9x0\u662f\u53ef\u5fae\u7684\u3002 A\u0394x\u53eb\u505a\u51fd\u6570\u5728\u70b9x0\u76f8\u5e94\u4e8e\u81ea\u53d8\u91cf\u589e\u91cf\u25b3x\u7684\u5fae\u5206\uff0c\u8bb0\u4f5cdy\uff0c\u5373\uff1ady=A\u0394x\u3002
\u5fae\u5206dy\u662f\u81ea\u53d8\u91cf\u6539\u53d8\u91cf\u25b3x\u7684\u7ebf\u6027\u51fd\u6570\uff0cdy\u4e0e\u25b3y\u7684\u5dee\u662f\u5173\u4e8e\u25b3x\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f\u91cf\uff0c\u6211\u4eec\u628ady\u79f0\u4f5c\u25b3y\u7684\u7ebf\u6027\u4e3b\u90e8\u3002\u5f97\u51fa\uff1a \u5f53\u25b3x\u21920\u65f6\uff0c\u25b3y\u2248dy\u3002
\u5bfc\u6570\u7684\u8bb0\u53f7\u4e3a\uff1a(dy)/(dx)=f\u2032(X)\uff0c\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0\uff0c\u5b83\u4e0d\u4ec5\u8868\u793a\u5bfc\u6570\u7684\u8bb0\u53f7\uff0c\u800c\u4e14\u8fd8\u53ef\u4ee5\u8868\u793a\u4e24\u4e2a\u5fae\u5206\u7684\u6bd4\u503c\uff08\u628a\u25b3x\u770b\u6210dx\uff0c\u5373\uff1a\u5b9a\u4e49\u81ea\u53d8\u91cf\u7684\u589e\u91cf\u7b49\u4e8e\u81ea\u53d8\u91cf\u7684\u5fae\u5206\uff09\uff0c\u8fd8\u53ef\u8868\u793a\u4e3ady=f\u2032(X)dX\u3002
y=arcsin\u221a(1-x^2)
y'=-x/(|x|\u221a(1-x^2))
\u2234dy=-xdx/(|x|\u221a(1-x^2))
\u5f53x>0 dy=-dx/\u221a(1-x^2)
\u5f53x<0 dy=dx/\u221a(1-x^2)
绛旓細:y=arcsin鈭1-x^2锛歞y/dx =1/鈭(1-(鈭(1-x^2)^2)) * (-x)/鈭(1-x^2)=1/|x| * (-x)/鈭(1-x^2)=-x/|x| * 鈭(1-x^2)
绛旓細鍑芥暟y=sinx鐨勫弽鍑芥暟鍙仛鍙嶆寮鍑芥暟锛岃浣渪=arcsiny.涔犳儻涓婄敤x琛ㄧず鑷彉閲忥紝鐢▂琛ㄧず鍑芥暟锛屾墍浠ュ弽姝ｅ鸡鍑芥暟鍐欐垚y=arcsinx.瀹氫箟鍩熸槸 [-1锛1] 锛屽煎煙鏄痽鈭 [锛嵪/2 , 蟺/2] ;閭ｄ箞瀵逛簬鏈鑰岃█锛屾牴鎹鍚堝嚱鏁拌姹 鍒欐湁-1鈮1-x^2鈮1鍗0鈮^2鈮2 鍒欏畾涔夊煙涓篬-鈭2,鈭2]...
绛旓細锛5锛塽锛X锛2 v锛1-u w锛濃垰v y锛漚rcsinw
绛旓細dy/dx =1/鈭(1-(鈭(1-x^2)^2)) * (-x)/鈭(1-x^2) =1/x * (-x)/鈭(1-x^2) =-x/x * 鈭(1-x^2)
绛旓細澶嶅悎鍑芥暟涓 y=arcsin(1-x^2)鐢-1<=1-x^2<=1锛屽緱0<=x^2<=2 鎵浠-鈭2<=x<=鈭2.鎵浠ュ嚱鏁扮殑瀹氫箟鍩熶负[-鈭2,鈭2]
绛旓細澶嶅悎鍑芥暟涓 y=arcsin(1-x^2) 鐢-1<=1-x^2<=1锛屽緱0<=x^2<=2 鎵浠-鈭2<=x<=鈭2. 鎵浠ュ嚱鏁扮殑瀹氫箟鍩熶负[-鈭2,鈭2]
绛旓細-1鈮(1-x^2)/(1+x^2)鈮1锛-1-x^2鈮1-x^2鈮1+x^2锛屾亽鎴愮珛锛屾墍浠鈭圧
绛旓細澶嶅悎鍑芥暟涓 y=arcsin(1-x^2)鐢-1<=1-x^2<=1锛屽緱0<=x^2<=2 鎵浠-鈭2<=x<=鈭2.鎵浠ュ嚱鏁扮殑瀹氫箟鍩熶负[-鈭2,鈭2]
绛旓細.y=arcsinx y'=1/鈭1-x^2 y'=(arcsin锛1-2x锛)'=1/鈭1-(1-2x)^2 =1/2鈭(x-x^2)
绛旓細1.) 瀹氫箟鍩熼』婊¤冻锛-1=<(1-x)^2<=1, 鍗 0=<x<=2 寰楀畾涔夊煙涓猴細 0=<x<=2 2)瀹氫箟鍩熼』婊¤冻锛歝os2x>=0, 鍗 2k蟺-蟺/2=<2x<=2k蟺+蟺/2, k 涓轰换鎰忔暣鏁 鍗 k蟺-蟺/4=<x<=k蟺+蟺/4
