散文網(wǎng) » 科技 »學(xué)習(xí) » 導(dǎo)數(shù)公式推導(dǎo)

導(dǎo)數(shù)公式推導(dǎo)

2023-07-08 22:01 作者:源來是小白 0人讀過 | 我要投稿

推導(dǎo)公式 $(x%5En)'%20%3D%20nx%5E%7Bn-1%7D$

設(shè)函數(shù) $f(x)%20%3D%20x%5En$

則該函數(shù)的導(dǎo)函數(shù):

$f'(x)%3D%5Clim_%7Bx%5Cto0%7D%5Cfrac%7B%5CDelta%20y%20%7D%7B%5CDelta%20x%7D%20%3D%5Clim_%7Bx%5Cto0%7D%5Cfrac%7Bf(x%2B%5CDelta%20x)%20-%20f(x)%7D%7B%5CDelta%20x%7D%20$

??? ?????? $%3D%5Clim_%7Bx%5Cto0%7D%5Cfrac%7B(x%2B%5CDelta%20x)%5En%20-%20x%5En%7D%7B%5CDelta%20x%7D%20$

?????????? $%3D%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7B1%7D%7B%5CDelta%20x%7D%20%5Csum_%7Br%20%3D%200%7D%5En%20C_%7Bn%7D%5Erx%5E%7Bn-r%7D%5CDelta%20x%20-%20x%5En%20%0A$

?? ?????? $%3D%5Clim_%7Bx%5Cto0%7D%5Cfrac%7B1%7D%7B%5CDelta%20x%7D(C_%7Bn%7D%5E0x%5En%2BC_%7Bn%7D%5E1x%5E%7Bn-1%7D%5CDelta%20x%2B%20...%2BC_%7Bn%7D%5E%7Bn-1%7Dx%5CDelta%20x%5E%7Bn-1%7D%2BC_%7Bn%7D%5En%5CDelta%20x%5En%20-%20x%5En)%20$

這里可以通過楊輝三角的規(guī)律來確定出:

???????????? $C_%7Bn%7D%5E0%3DC_%7Bn%7D%5En%3D1$

???????????? $C_%7Bn%7D%5E1%3DC_%7Bn%7D%5E%7Bn-1%7D%3Dn$

帶入這幾個值, 得:

????? ? ? ????? $%5Clim_%7Bx%5Cto0%7D%5Cfrac%7B1%7D%7B%5CDelta%20x%7D(x%5En%2Bnx%5E%7Bn-1%7D%5CDelta%20x%2B%20...%2Bnx%5CDelta%20x%5E%7Bn-1%7D%2B%5CDelta%20x%5En%20-%20x%5En)%20$

?????????? $%3D%5Clim_%7Bx%5Cto0%7D%5Cfrac%7B1%7D%7B%5CDelta%20x%7D(nx%5E%7Bn-1%7D%5CDelta%20x%2B%20...%2Bnx%5CDelta%20x%5E%7Bn-1%7D%2B%5CDelta%20x%5En)%20$

?? ??????? $%3D%5Clim_%7Bx%5Cto0%7Dnx%5E%7Bn-1%7D%2B%20nx%5E%7Bn-2%7D%5CDelta%20x%2B...%2Bnx%5CDelta%20x%5E%7Bn-2%7D%2B%5CDelta%20x%5E%7Bn-1%7D$

?? ?? ? ?? $%20%5Crightarrow%20nx%5E%7Bn-1%7D%20%E5%BD%93%5CDelta%20x%20%5Crightarrow%200%E6%97%B6%5C$

∴ $(x%5En)'%20%3D%20nx%5E%7Bn-1%7D$

類似的, 我們也可以推導(dǎo)一下函數(shù) $f(x)%20%3D%20ax%5E2%2Bbx%2Bc$ 的導(dǎo)函數(shù):

$f'(x)%20%3D%20%5Clim_%7Bx%5Cto0%7D%5Cfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%20%20%3D%20%5Clim_%7Bx%5Cto0%7D%5Cfrac%7Bf(x%2B%5CDelta%20x)-f(x)%7D%7B%5CDelta%20x%7D%20$