莱布尼茨公式求高阶导数？ 用莱布尼茨公式求高阶导数（题简单，过程不太会）
\u9ad8\u7b49\u6570\u5b66\u9ad8\u9636\u5bfc\u6570\u83b1\u5e03\u5c3c\u5179\u516c\u5f0f\u83b1\u5e03\u5c3c\u5179\u516c\u5f0f\u597d\u6bd4\u4e8c\u9879\u5f0f\u5b9a\u7406\uff0c\u5b83\u662f\u7528\u6765\u6c42f(x)*g(x)\u7684\u9ad8\u9636\u5bfc\u6570\u7684\u3002
(uv)' = u'v+uv'\uff0c
(uv)'\u2018 = u'\u2019v+2u'v'+uv'\u2018
\u4f9d\u6570\u5b66\u5f52\u7eb3\u6cd5\uff0c\u2026\u2026\uff0c\u53ef\u8bc1\u8be5\u83b1\u5e03\u5c3c\u5179\u516c\u5f0f\u3002
\u5404\u4e2a\u7b26\u53f7\u7684\u610f\u4e49
\u03a3--------------\u6c42\u548c\u7b26\u53f7
C(n,k)--------\u7ec4\u5408\u7b26\u53f7\uff0c\u5373n\u53d6k\u7684\u7ec4\u5408
u^(n-k)-------u\u7684n-k\u9636\u5bfc\u6570
v^(k)----------v\u7684k\u9636\u5bfc\u6570
\u8fd9\u4e2a\u516c\u5f0f\u548c\u6392\u5217\u7ec4\u5408\u4e2d\u7684\u4e8c\u9879\u5f0f\u5b9a\u7406\u76f8\u4f3c\uff0c\u4e8c\u9879\u5f0f\u5b9a\u7406\u4e2d\u7684\u591a\u5c11\u6b21\u65b9\u5728\u8fd9\u91cc\u6539\u4e3a\u591a\u5c11\u9636\u5bfc\u6570\u3002
\uff08uv\uff09\u4e00\u9636\u5bfc=u\u4e00\u9636\u5bfc\u4e58\u4ee5v+u\u4e58\u4ee5v\u4e00\u9636\u5bfc
\uff08uv\uff09\u4e8c\u9636\u5bfc=u\u4e8c\u9636\u5bfc\u4e58\u4ee5v+2\u500du\u4e00\u9636\u5bfc\u4e58\u4ee5v\u4e00\u9636\u5bfc+u\u4e58\u4ee5v\u4e8c\u9636\u5bfc
\uff08uv\uff09\u4e09\u9636\u5bfc=u\u4e09\u9636\u5bfc\u4e58\u4ee5v+3\u500du\u4e8c\u9636\u5bfc\u4e58\u4ee5v\u4e00\u9636\u5bfc+3\u500du\u4e00\u9636\u5bfc\u4e58\u4ee5v\u4e8c\u9636\u5bfc+u\u4e58\u4ee5v\u4e09\u9636\u5bfc
\u6269\u5c55\u8d44\u6599\uff1a\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\u7684\u63a8\u5bfc\u8fc7\u7a0b
\u5982\u679c\u5b58\u5728\u51fd\u6570u=u(x)\u4e0ev=v(x)\uff0c\u4e14\u5b83\u4eec\u5728\u70b9x\u5904\u90fd\u5177\u6709n\u9636\u5bfc\u6570\uff0c\u90a3\u4e48\u663e\u800c\u6613\u89c1\u7684\uff0c
u(x) \u00b1 v(x) \u5728x\u5904\u4e5f\u5177\u6709n\u9636\u5bfc\u6570\uff0c\u4e14 (u\u00b1v)(n)= u(n)\u00b1 v(n)
\u81f3\u4e8eu(x) \u00d7 v(x) \u7684n\u9636\u5bfc\u6570\u5219\u8f83\u4e3a\u590d\u6742\uff0c\u6309\u7167\u57fa\u672c\u6c42\u5bfc\u6cd5\u5219\u548c\u516c\u5f0f\uff0c\u53ef\u4ee5\u5f97\u5230\uff1a
(uv)' = u'v + uv'
(uv)'' = u''v + 2u'v' + uv''
(uv)''' = u'''v + 3u''v' + 3u'v'' + uv'''
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f
\u5728x=0\u7684\u65f6\u5019
\u53ea\u6709\u5bf9x²\u6c42\u5bfc\u4e24\u6b21\u65f6\uff0c\u6574\u4e2a\u5f0f\u5b50\u7684\u5bfc\u6570\u624d\u4e0d\u7b49\u4e8e0
\u5373\u5bf92^x\u6c42\u5bfcn-2\u6b21
\u9996\u5148C(n,2)*2=n(n-1)
\u800c\u8fd9\u91cc\u7684(2^x)(n-2)\uff0cn-2\u4e3a\u4e0a\u6807
\u6307\u7684\u662f\u5bf92^x\u6c42\u5bfcn-2\u6b21
\u663e\u71362^x\u5bfc\u6570\u4e3aln2 *2^x
\u90a3\u4e48n-2\u9636\u5bfc\u6570\u5c31\u662f(ln2)^(n-2) *2^x
\u4e8e\u662f\u518d\u4e58\u4ee5C(n,2)*2\u5373n(n-1)
\u5176n\u9636\u5bfc\u6570\u4e3an(n-1) *(ln2)^(n-2)
\u4ece(uv)' = u'v+uv'\uff0c
(uv)'\u2018 = u'\u2019v+2u'v'+uv'\u2018\uff0c
\u4f9d\u6570\u5b66\u5f52\u7eb3\u6cd5\uff0c\u53ef\u8bc1\u8be5\u83b1\u5e03\u5c3c\u5179\u516c\u5f0f\u3002
\u5f04\u61c2\u5404\u4e2a\u7b26\u53f7\u7684\u610f\u4e49\uff0c\u4f1a\u4f7f\u7528\u5c31\u884c\u4e86\uff1a
\u03a3--------------\u6c42\u548c\u7b26\u53f7\uff1b
C(n,k)--------\u7ec4\u5408\u7b26\u53f7\uff0c\u5373n\u53d6k\u7684\u7ec4\u5408\uff1b
u^(n-k)-------u\u7684n-k\u9636\u5bfc\u6570\uff1b
v^(k)----------v\u7684k\u9636\u5bfc\u6570\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u5982\u679c\u5b58\u5728\u51fd\u6570u=u(x)\u4e0ev=v(x)\uff0c\u4e14\u5b83\u4eec\u5728\u70b9x\u5904\u90fd\u5177\u6709n\u9636\u5bfc\u6570\uff0c\u90a3\u4e48\u663e\u800c\u6613\u89c1\u7684\uff0c
u(x) \u00b1 v(x) \u5728x\u5904\u4e5f\u5177\u6709n\u9636\u5bfc\u6570\uff0c\u4e14 (u\u00b1v)(n)= u(n)\u00b1 v(n)
\u81f3\u4e8eu(x) \u00d7 v(x) \u7684n\u9636\u5bfc\u6570\u5219\u8f83\u4e3a\u590d\u6742\uff0c\u6309\u7167\u57fa\u672c\u6c42\u5bfc\u6cd5\u5219\u548c\u516c\u5f0f\uff0c\u53ef\u4ee5\u5f97\u5230\uff1a
(uv)' = u'v + uv'
(uv)'' = u''v + 2u'v' + uv''
(uv)''' = u'''v + 3u''v' + 3u'v'' + uv'''
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f
只有对x²求导两次时，整个式子的导数才不等于0
即对2^x求导n-2次
首先C(n,2)*2=n(n-1)
而这里的(2^x)(n-2)，n-2为上标
指的是对2^x求导n-2次
显然2^x导数为ln2 *2^x
那么n-2阶导数就是(ln2)^(n-2) *2^x
于是再乘以C(n,2)*2即n(n-1)
其n阶导数为n(n-1) *(ln2)^(n-2)
绛旓細姹傞珮闃跺鏁锛氬彟澶栨柟娉曟眰涔嬶細搴旂敤绉殑瀵兼暟鍏紡閫愰樁姹傚锛屽鏋滃墠3銆5闃跺鏁拌兘鐪嬪嚭瑙勫緥鏈濂
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