极坐标求导 这个极坐标的导数怎么求？
\u5982\u4f55\u6c42\u6781\u5750\u6807\u7684\u5bfc\u6570\u53ef\u4ee5\u8f6c\u5316\u6210\u76f4\u89d2\u5750\u6807\u7cfb\u8fdb\u884c\u6c42\u5bfc\u3002
\u6781\u5750\u6807\u65b9\u7a0b\u6709\u4e24\u4e2a\u53c2\u6570\uff1a\u6a21\u957fr\u548c\u8f90\u89d2t\uff0c\u8fd8\u53ef\u4ee5\u5bf9\u6781\u5750\u6807\u65b9\u7a0br=r(t)\u6c42\u5bfc\uff0c\u5c31\u548c\u5728\u76f4\u89d2\u5750\u6807\u7cfb\u4e2d\u6c42\u5bfc\u7684\u8fc7\u7a0b\u53ca\u65b9\u6cd5\u90fd\u4e00\u6837\uff0c\u5373r\u5bf9t\u6c42\u5bfc\u3002
\u53ea\u662f\u8fd9\u4e2a\u5bfc\u6570\u7684\u542b\u4e49\u6709\u6240\u4e0d\u540c\uff0c\u662f\u6307\u6a21\u957fr\u5173\u4e8e\u8f90\u89d2t\u7684\u53d8\u5316\u7387\u3002
\u51e0\u4f55\u610f\u4e49
\u7528\u6781\u5750\u6807\u89e3\u51b3\u51e0\u4f55\u95ee\u9898\u7684\u65b9\u6cd5\u3002\u5728\u76f4\u89d2\u5750\u6807\u7cfb\u4e2d\uff08x,y\uff09\uff0cx\u88ab\u03c1cos\u03b8\u4ee3\u66ff\uff0cy\u88ab\u03c1sin\u03b8\u4ee3\u66ff\uff0c\u03c1=(x^2+y^2)^0.5,\u4ece\u800c\u5f97\u5230\u65b0\u7684\u65b9\u7a0b\u3002\u8fd9\u6837\u7684\u65b9\u7a0b\u5e38\u5e38\u7528\u6765\u89e3\u51b3\u66f2\u7ebf\u95ee\u9898\uff0c\u5982\u692d\u5706\u66f2\u7ebf\u3001\u7ebd\u7ebf\u3001\u87ba\u7ebf\u7b49\u7b49\uff0c\u53ef\u4ee5\u4f7f\u89e3\u9898\u66f4\u52a0\u6e05\u6670\u7b80\u4fbf\u3002
\u8bbe\u66f2\u7ebfC\u7684\u6781\u5750\u6807\u65b9\u7a0b\u4e3ar=r\uff08\u03b8\uff09\u3002
\u5219C\u7684\u53c2\u6570\u65b9\u7a0b\u4e3a{ x=r\uff08\u03b8\uff09cos\u03b8
y=r\uff08\u03b8\uff09sin\u03b8
\u5c31\u662f\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc\u554a\uff0c
x=acos\u03b8^3
y=asin\u03b8^3
dy/dx=(dy/d\u03b8)*(d\u03b8/dx)=3asin\u03b8^2cos\u03b8/-3acos\u03b8^2sin\u03b8=-sin\u03b8/cos\u03b8,
\u4e8c\u9636\u5bfc\u9053\u7406\u4e00\u6837\u7684\u5c31\u662f\u8ba1\u7b97\u9ebb\u70e6
绛旓細鏋佸潗鏍鏂圭▼鐨勬洸鐜囧叕寮忥細k=(d²r/d胃²)/(1-r²)
绛旓細x琚乧os胃浠ｆ浛锛寉琚乻in胃浠ｆ浛锛屜=鈭(x^2+y^2)浠庤屽緱鍒版柊鐨勬柟绋嬨傝繖鏍风殑鏂圭▼甯稿父鐢ㄦ潵瑙ｅ喅鏇茬嚎闂锛屽妞渾鏇茬嚎銆佺航绾裤佽灪绾跨瓑绛夛紝鍙互浣胯В棰樻洿鍔犳竻鏅扮畝渚裤傝鏇茬嚎C鐨鏋佸潗鏍鏂圭▼涓簉=r锛埼革級銆傚垯C鐨勫弬鏁版柟绋嬩负{ x=r锛埼革級cos胃y=r锛埼革級sin胃鍏朵腑胃涓烘瀬瑙掋傜敱鍙傛暟鏂圭▼姹傚娉曪紝寰楁洸绾緾鐨...
绛旓細r=ae^(m@)鍏堝寲涓虹洿瑙鍧愭爣 x=ae^(m@)*cos y=ae^(m@)*sin 鐒跺悗鎶婂弬鏁版柟绋姹傚 dy/dx =(dy/d@)/(dx/d@)=(ae^(m@)*sin@)'/(ae^(m@)*cos@)'=[a(e^(m@)*m*sin@+e^(m@)*cos@]/[a(e^(m@)*m*cos@+e^(m@)*(-sin@)]=(msin@+cos@)/(mcos@-sin@)=(mtan...
绛旓細鍦鏋佸潗鏍涓紝x琚乧os胃浠ｆ浛锛寉琚乻in胃浠ｆ浛銆傁乛2=(x^2+y^2)鏋佸潗鏍囩郴涓殑涓や釜鍧愭爣 r 鍜 胃 鍙互鐢变笅闈㈢殑鍏紡杞崲涓虹洿瑙掑潗鏍囩郴涓嬬殑鍧愭爣鍊 x = r*cos锛埼革級锛寉 = r*sin锛埼革級锛岀敱涓婅堪浜屽叕寮忥紝鍙緱鍒颁粠鐩磋鍧愭爣绯讳腑x 鍜 y 涓ゅ潗鏍囧浣曡绠楀嚭鏋佸潗鏍囦笅鐨勫潗鏍 r = sqrt(x^2 + y^...
绛旓細鍋忓鏁扮鍙锋槸琛ㄧず涓涓暣浣擄紝涓嶈兘鎶婂畠鍒嗗紑鏉ョ湅锛屽畠璺熶竴鍏冨嚱鏁扮殑瀵兼暟绗﹀彿涓嶄竴鏍枫傛渶鍚庝竴姝ョ殑涔樻硶涓嶈兘寰楀埌缁撴灉涓1銆傗埖x=rcos胃,y=rsin胃 鈭磖=鈭(x²+y²)鍒檇r/dx=2x/2鈭(x²+y²) =x/r=rcos胃/r=cos胃 dr/dy=2y/2鈭(x²+y²) =y/r=rsin胃/r=...
绛旓細鏋佸潗鏍r(胃)鍑芥暟琛ㄧず鐨勫氨鏄 鍗婂緞r涓庤搴ξ镐箣闂寸殑鍏崇郴 閭ｄ箞鐜板湪r瀵刮姹傚 寰楀埌鐨勫鏁板綋鐒跺氨鏄崐寰剅 鍦ㄦ煇胃鍊兼椂鐨勫彉鍖栫巼
绛旓細r = a(1+cost) 寰 r^2 = ar(1+cost) = ar +arcost 鍒 x^2+y^2 = a鈭(x^2+y^2) + ax 涓よ竟瀵 x 姹傚锛屽緱 2x+2yy' = a(x+yy')/鈭(x^2+y^2) + a 2x鈭(x^2+y^2)+2yy'鈭(x^2+y^2) = ax+ayy'+a鈭(x^2+y^2)y' = [ax+a鈭(x^2+y^2)-2x鈭...
绛旓細鍙互鎯虫垚姣忎釜寰皬瑙掑害鐨勫渾闈㈢Н涔嬪拰锛屽渾鐨勯潰绉槸浜屽垎涔嬩竴涔樹互鍗婂緞鐨勫钩鏂逛箻浠ヨ搴ο锛屾墍浠ヨ繖閲屾槸r鐨勫钩鏂
绛旓細棣栧厛浣犵殑缁撴灉鑲畾鏄笉瀵圭殑锛屽洜涓哄鏋溛哥殑鍏ㄥ井鍒嗚〃杈惧紡鏄-sin胃d胃=1/rdx鐨勮瘽锛岀敱浜庤繖閲屾病鏈塪y椤癸紝鏁呂稿y鐨勫亸瀵兼暟绛変簬0锛岃繖鏄剧劧鏄笉鍙兘鐨勩傞敊璇師鍥犲湪浜庡鏋滄妸x锛寉鐪嬪仛鍙橀噺锛屾妸胃鐪嬪仛x,y鐨勫嚱鏁扮殑璇濓紝鍒檙=鈭(x^2+y^2)涔熸槸鍏充簬x,y鐨勫嚱鏁帮紝姹傚鏃秗搴旇褰撳仛涓棿鍙橀噺澶勭悊锛岃屼笉鑳芥妸r鐪嬫垚...
绛旓細楂樻暟甯歌鍑芥暟姹傚鍏紡濡備笅鍥撅細姹傚鏄暟瀛﹁绠椾腑鐨勪竴涓绠楁柟娉曪紝瀹冪殑瀹氫箟灏辨槸锛屽綋鑷彉閲忕殑澧為噺瓒嬩簬闆舵椂锛屽洜鍙橀噺鐨勫閲忎笌鑷彉閲忕殑澧為噺涔嬪晢鐨勬瀬闄愩傚湪涓涓嚱鏁板瓨鍦ㄥ鏁版椂锛岀О杩欎釜鍑芥暟鍙鎴栬呭彲寰垎銆傚彲瀵肩殑鍑芥暟涓瀹氳繛缁備笉杩炵画鐨勫嚱鏁颁竴瀹氫笉鍙銆備竴闃跺鏁扮殑鍙樺寲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...
