导数八个公式和运算法则 导数八个公式和运算法则
\u5bfc\u6570\u516b\u4e2a\u516c\u5f0f\u548c\u8fd0\u7b97\u6cd5\u5219\u662f\u4ec0\u4e48\uff1f\u516b\u4e2a\u516c\u5f0f\uff1ay=c(c\u4e3a\u5e38\u6570) y'=0\uff1by=x^n y'=nx^(n-1)\uff1by=a^x y'=a^xlna y=e^x y'=e^x\uff1by=logax y'=logae/x y=lnx y'=1/x \uff1by=sinx y'=cosx \uff1by=cosx y'=-sinx \uff1by=tanx y'=1/cos^2x \uff1by=cotx y'=-1/sin^2x\u3002
\u8fd0\u7b97\u6cd5\u5219\uff1a
\u52a0\uff08\u51cf\uff09\u6cd5\u5219\uff1a[f(x)+g(x)]'=f(x)'+g(x)'
\u4e58\u6cd5\u6cd5\u5219\uff1a[f(x)*g(x)]'=f(x)'*g(x)+g(x)'*f(x)
\u9664\u6cd5\u6cd5\u5219\uff1a[f(x)/g(x)]'=[f(x)'*g(x)-g(x)'*f(x)]/g(x)^2
\u4e00\u4e2a\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u7684\u5bfc\u6570\u63cf\u8ff0\u4e86\u8fd9\u4e2a\u51fd\u6570\u5728\u8fd9\u4e00\u70b9\u9644\u8fd1\u7684\u53d8\u5316\u7387\u3002\u5982\u679c\u51fd\u6570\u7684\u81ea\u53d8\u91cf\u548c\u53d6\u503c\u90fd\u662f\u5b9e\u6570\u7684\u8bdd\uff0c\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u7684\u5bfc\u6570\u5c31\u662f\u8be5\u51fd\u6570\u6240\u4ee3\u8868\u7684\u66f2\u7ebf\u5728\u8fd9\u4e00\u70b9\u4e0a\u7684\u5207\u7ebf\u659c\u7387\u3002
\u901a\u8fc7\u6781\u9650\u7684\u6982\u5ff5\u5bf9\u51fd\u6570\u8fdb\u884c\u5c40\u90e8\u7684\u7ebf\u6027\u903c\u8fd1\u3002\u4f8b\u5982\u5728\u8fd0\u52a8\u5b66\u4e2d\uff0c\u7269\u4f53\u7684\u4f4d\u79fb\u5bf9\u4e8e\u65f6\u95f4\u7684\u5bfc\u6570\u5c31\u662f\u7269\u4f53\u7684\u77ac\u65f6\u901f\u5ea6\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\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
\u51fd\u6570y=f\uff08x\uff09\u5728x0\u70b9\u7684\u5bfc\u6570f'\uff08x0\uff09\u7684\u51e0\u4f55\u610f\u4e49\uff1a\u8868\u793a\u51fd\u6570\u66f2\u7ebf\u5728\u70b9P0\uff08x0,f\uff08x0\uff09\uff09\u5904\u7684\u5207\u7ebf\u7684\u659c\u7387\uff08\u5bfc\u6570\u7684\u51e0\u4f55\u610f\u4e49\u662f\u8be5\u51fd\u6570\u66f2\u7ebf\u5728\u8fd9\u4e00\u70b9\u4e0a\u7684\u5207\u7ebf\u659c\u7387\uff09\u3002
\u82e5\u5bfc\u6570\u5927\u4e8e\u96f6\uff0c\u5219\u5355\u8c03\u9012\u589e\uff1b\u82e5\u5bfc\u6570\u5c0f\u4e8e\u96f6\uff0c\u5219\u5355\u8c03\u9012\u51cf\uff1b\u5bfc\u6570\u7b49\u4e8e\u96f6\u4e3a\u51fd\u6570\u9a7b\u70b9\uff0c\u4e0d\u4e00\u5b9a\u4e3a\u6781\u503c\u70b9\u3002\u9700\u4ee3\u5165\u9a7b\u70b9\u5de6\u53f3\u4e24\u8fb9\u7684\u6570\u503c\u6c42\u5bfc\u6570\u6b63\u8d1f\u5224\u65ad\u5355\u8c03\u6027\u3002
\u82e5\u5df2\u77e5\u51fd\u6570\u4e3a\u9012\u589e\u51fd\u6570\uff0c\u5219\u5bfc\u6570\u5927\u4e8e\u7b49\u4e8e\u96f6\uff1b\u82e5\u5df2\u77e5\u51fd\u6570\u4e3a\u9012\u51cf\u51fd\u6570\uff0c\u5219\u5bfc\u6570\u5c0f\u4e8e\u7b49\u4e8e\u96f6\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1\u2014\u2014\u5bfc\u6570
(x^n)'
=
(x+deta
x)^n
-
x^n\uff0c\u4e00\u9636\u5bfc\u6570\u53d6\u4e00\u9636\u65e0\u7a77\u5c0f\u91cf\uff0c\u5373ndetax*x^(n-1),\u9664\u4ee5detax\u5373\u4e3a\u5176\u5bfc\u6570\uff1b
3\u30014\u4f9d\u4e0a\u5904\u7406\uff0c\u4f46\u8981\u5c06\u6307\u6570\u548c\u5bf9\u6570\u9879\u8fdb\u884c\u6cf0\u52d2\u5c55\u5f00\uff0c\u7136\u540e\u518d\u8fd0\u7b97\uff1b
\u56db\u5219\u8fd0\u7b97\u662f\u7531\u5b9e\u6570\u96c6\u7684\u6027\u8d28\u51b3\u5b9a\u7684\uff0c\u6ca1\u4ec0\u4e48\u597d\u8bf4\u7684
\u8be6\u7ec6\u5199\u6765\u592a\u9ebb\u70e6\u4e86\uff0c\u81ea\u5df1\u641c\u7d22\u4e00\u4e0b\uff0c\u6216\u8005\u4e13\u95e8\u627e\u672c\u8bb2\u5bfc\u6570\u548c\u5fae\u5206\u7684\u6570\u5b66\u4e66\u770b\u5427
八个公式：
1.y=c(c为常数) y'=0
2.y=x^n y'=nx^(n-1)
3.y=a^x y'=a^xlna
y=e^x y'=e^x
4.y=logax y'=logae/x y=lnx y'=1/x
5.y=sinx y'=cosx
6.y=cosx y'=-sinx
7.y=tanx y'=1/cos^2x
8.y=cotx y'=-1/sin^2x
运算法则：
加（减）法则：[f(x)+g(x)]'=f(x)'+g(x)'
乘法法则：[f(x)*g(x)]'=f(x)'*g(x)+g(x)'*f(x)
除法法则：[f(x)/g(x)]'=[f(x)'*g(x)-g(x)'*f(x)]/g(x)^2
1.y=c(c为常数) y'=0
2.y=x^n y'=nx^(n-1)
3.y=a^x y'=a^xlna
y=e^x y'=e^x
4.y=logax y'=logae/x
y=lnx y'=1/x
5.y=sinx y'=cosx
6.y=cosx y'=-sinx
7.y=tanx y'=1/cos^2x
8.y=cotx y'=-1/sin^2x
.加（减）法则：[f(x)+g(x)]'=f(x)'+g(x)'
乘法法则： [f(x)*g(x)]'=f(x)'*g(x)+g(x)'*f(x)
除法法则：[f(x)/g(x)]'=[f(x)'*g(x)-g(x)'*f(x)]/g(x)^2
绛旓細鍏釜鍏紡锛y=c(c涓哄父鏁) y'=0锛泍=x^n y'=nx^(n-1)锛泍=a^x y'=a^xlna y=e^x y'=e^x锛泍=logax y'=logae/x y=lnx y'=1/x 锛泍=sinx y'=cosx 锛泍=cosx y'=-sinx 锛泍=tanx y'=1/cos^2x 锛泍=cotx y'=-1/sin^2x銆杩愮畻娉曞垯锛氬姞锛堝噺锛娉曞垯锛歔f(x)+g(x)]'...
绛旓細鍏釜鍏紡锛y=c(c涓哄父鏁) y'=0锛泍=x^n y'=nx^(n-1)锛泍=a^x y'=a^xlna y=e^x y'=e^x锛泍=logax y'=logae/x y=lnx y'=1/x 锛泍=sinx y'=cosx 锛泍=cosx y'=-sinx 锛泍=tanx y'=1/cos^2x 锛泍=cotx y'=-1/sin^2x銆杩愮畻娉曞垯锛氬姞锛堝噺锛娉曞垯锛歔f(x)+g(x)]'...
绛旓細瀵兼暟鍏紡锛1.y=c(c涓哄父鏁)y'=0 2.y=x^n y'=nx"(n-1)3.y=a^x y'=a xlna y=e^x y'=e^x 4.y=logax y'=logae/x y=lnx y'=1/x 5.y=sinx y'=cosx 6.y=cosx y'=-sinx 7.y=tanx y'=1/cos~2x 8.y=cotx y'=-1/sin^2x 杩愮畻娉曞垯锛氬噺娉曟硶鍒欙細(f(x)涓g(x)...
绛旓細瀵兼暟鍏紡 1.y=c(c涓哄父鏁)y=02.y=x^ny=nx^(n-1)3.y=a^xy=a^xlnay=e^xy=e^x4.y=logaxy=logae/xy=lnxy=1/x5.y=sinxy=cosx6.y=cosxy=-sinx7.y=tanxy=1/cos^2x8.y=cotxy=-1/sin^2x 杩愮畻娉曞垯 鍑忔硶娉曞垯锛(f(x)-g(x))=f(x)-g(x)鍔犳硶娉曞垯锛(f(x)+g(x))=f(...
绛旓細瀵兼暟鍏紡锛歽=c(c涓哄父鏁) y'=0銆亂=x^n y'=nx^(n-1) 锛杩愮畻娉曞垯锛氬姞锛堝噺锛娉曞垯锛歔f(x)+g(x)]'=f(x)'+g(x)'銆傚鏁板叕寮 1.y=c(c涓哄父鏁) y'=0 2.y=x^n y'=nx^(n-1)3.y=a^x y'=a^xlna y=e^x y'=e^x 4.y=logax y'=logae/x y=lnx y'=1/x 5.y...
绛旓細杩愮畻娉曞垯 鍑忔硶娉曞垯锛(f(x)-g(x))'=f'(x)-g'(x)鍔犳硶娉曞垯锛(f(x)+g(x))'=f'(x)+g'(x)涔樻硶娉曞垯锛(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)闄ゆ硶娉曞垯锛(g(x)/f(x))'=(g'(x)f(x)-f'(x)g(x))/(f(x))^2 ...
绛旓細涓瀵间箻浜+涓涔樹簩瀵硷紙鍗斥憽寮忥級銆3銆佷袱涓嚱鏁扮殑鍟嗙殑瀵煎嚱鏁涔熸槸涓涓垎寮忥細锛堝瓙瀵间箻姣-瀛愪箻姣嶅锛夐櫎浠ユ瘝骞虫柟锛堝嵆鈶㈠紡锛夈4銆佸鏋滄湁澶嶅悎鍑芥暟锛屽垯鐢ㄩ摼寮娉曞垯姹傚銆傞珮闃瀵兼暟鐨勬眰娉 1锛庣洿鎺ユ硶锛氱敱楂橀樁瀵兼暟鐨勫畾涔夐愭姹傞珮闃跺鏁般備竴鑸敤鏉ュ鎵捐В棰樻柟娉曘2锛庨珮闃跺鏁扮殑杩愮畻娉曞垯锛
绛旓細C鈥=0锛圕涓哄父鏁帮級锛坸鈭锛夆=nx鈭э紙n-1锛夛紙sinx锛夆=cosx 锛坈osx锛夆=-sinx 鍑芥暟鐨勫拰路宸风Н路鍟嗙殑瀵兼暟锛氾紙u卤v锛夆=u鈥猜眝鈥诧紙uv锛夆=u鈥瞯+uv鈥诧紙u/v锛夆=锛坲鈥瞯-uv鈥诧級/v²瀵兼暟 鏄嚱鏁扮殑灞閮ㄦц川銆備竴涓嚱鏁板湪鏌愪竴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鍙樺寲鐜囥傚鏋滃嚱鏁...
绛旓細瀵兼暟鍩烘湰鍏紡锛1. 瀵逛簬甯告暟c锛屽叾瀵兼暟涓0锛屽嵆y=c鐨勫鏁皔'=0銆2. 瀵逛簬骞傚嚱鏁皔=x^n锛屽叾瀵兼暟涓簄x^(n-1)锛屽嵆y'=nx^(n-1)銆瀵兼暟杩愮畻娉曞垯锛1. 瀵逛簬甯告暟c锛屽叾瀵兼暟涓0锛屽嵆y'=0銆2. 瀵逛簬骞傚嚱鏁皔=x^n锛屽叾瀵兼暟涓簄x^(n-1)锛屽嵆y'=nx^(n-1)銆3. 瀵逛簬鎸囨暟鍑芥暟y=a^x锛屽叾瀵兼暟涓篴^x*...
绛旓細鍏紡锛歽=c锛坈涓哄父鏁帮級y'=0銆亂=x^n y'=nx^锛坣-1锛夛紱杩愮畻娉曞垯锛氬姞锛堝噺锛夋硶鍒锛歔f锛坸锛+g锛坸锛塢'=f锛坸锛'+g锛坸锛'銆備竴涓嚱鏁板湪鏌愪竴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鍙樺寲鐜囥傚鏋滃嚱鏁扮殑鑷彉閲忓拰鍙栧奸兘鏄疄鏁扮殑璇濓紝鍑芥暟鍦ㄦ煇涓鐐圭殑瀵兼暟灏辨槸璇ュ嚱鏁版墍浠ｈ〃鐨勬洸绾垮湪杩欎竴鐐逛笂鐨勫垏绾挎枩鐜...
