第十次课 20环境工程一本40（56）人
① 基本初等函数的高阶导数的计算 ② 高阶导数的几个基本公式 ③ 隐函数的高阶导数
④ 参数确定的函数的高阶导数 ⑤ 微分的基本概念：
幂函数的高阶导数问题：
y?a0?a1x?a2x2??anxn
y?a0?a1x?a2x2?y??a1?2a2x?3a3x2?y????3!a3?4!a4x?................y?n??anxn?nanxn?1?n?n?1?anxn?2
y???2!a2?3!a3x?4?3a4x2??n?n?1??n?2?anxn?3y?k??0??n!an??ak??k!?n?幂函数的高阶导数
?1?????x??n???1??xn!?n?1n?1?????x?1???1?n!??n?1?x?1?n?1???1???2?x?x2?1?????1?2!???x3?x?3?1??????1?3!???x4?x?
?1?1?????2?x?1??x?1??1?????1?2!???3?x?1??x?1?2
?1????x?n?n???1??n!?1??????1?33!?xn?1??x?1?????x?1?4??n?1????x?1??n???1?n!?n?1?x?1?n?1?????x?a??n???1?n!??n?1?x?a?
?1????1?x??n??n!?1?x?n?1?1?????????b?x??n??n!?b?x?n?11?1?????2?1?x??1?x?2!?1??????3?1?x??1?x?3!?1???????4?1?x??1?x?对数函数的高阶导数：
?ln?1?x??????n?1??n?1?!??ln?1?x???n?x?1??n?n?1?n?1?!n?1?x??n?1????1ln1?x????1?x ln?1?x?1?ln?1?x????1?x???n??1??????1?x?n!n?1???ln?1?x??
?n??1?????1?x??n?1???1??n?1?!?n?x?1?n?1?n?1?!
n?1?x??1????1?x??n???1?x?几个常见的高阶导数公式： 1、?u?v?w?2、?Cu?3、?uv??n??n??u?n??v?n??w?n?
?Cu?n?
k?k??n?k?0?0??n?0?1?1??n?1???Cnuv?Cnuv?Cnuv?k?0n?n?
?uv??n?k?k??n?k?0?0??n?0?1?1??n?1???Cnuv?Cnuv?Cnuv?k?0nn?a?b?nkkn?k??Cnabk?0????????????????????????uv???u?v?uv??????????????????uv????u??v?2u?v??uv??????????uv?????u???v?3u??v??3u?v???uv???
?uv????u?4?v?4u???v??6u??v???4u?v????uv?4?4隐函数的高阶导数：
d2yx?y?2siny?0?,?2.dxdydy1??2cosy?0dxdx??dy1d2y?1??2?????2dx1?2cosydx1?2cosy??x?1?2cosy???2siny?2sinydydx?2siny1 1?2cosy2?1?2cosy??1?2cosy?3参数确定的函数的高阶导数：
张三的打字速度是李四的3倍，王五的打字速度是李四的4倍，张三的打字速度是王五的几倍？3/4.
?d2y?x???t???????2?????y??t??dx?????t??t???dt?t???t??t??t???t???t??t??t?1dy?t??t?d2y??t??t?? ?,??2?????????2????dx????dx?t?t?dx?t?t???t?t??x??t?t??tdxdt?t???t??t??t???t???t??t??t?1?t???t??t??t???t???t??t??t?????t?2?t????t??t?3?t?2??x?1?td3y??3?3??y?t?tdxdy1?3t2d2y?1?3t2??16t2?213t2?1?,2???????2dx?2tdx?4t3??2t?t?2t??2t??2t d3y?3t2?1??112t4?12t2112t4?12t23t2?3????????2753dx3??4t3?t?2t?2t?32t?8t?4t???x?1?t2?3?dy3t2?3?3??8t5?dx1?n????y??2x?5x?6111y????x?2??x?3?x?2x?3y?y?n?
nn?1?????x?2??n??1?????x?3??n???1?n!??1?n!??n?1n?1?x?2??x?3?微分概念：
y?f?x????x0??x?x,?????x?x?x0,???????????????????????????????????????????y?f?x??f?x0??????y?f?x0??x??f?x0????x?0lim?y?0???????c.tlim
?y????f??x0?????is?diff.?x?0?xf?x???on?x0?is?diff.?y?A?x????x??A?x????x????x???ylim?lim?lim?A???A?f??x0??x?0?x?x?0?x?0?x?x???y?yf??x0??lim??f??x0????x??????x??0??x?0?x?x?y?f??x0??x???x??x??f??x0??x????x??x?0
lim??x??x?x?lim??x??0?x?0dy?f??x0??x dx??x????????????????????????????
2021年高等数学课件-第十次课 20环境工程一本40
