A-0-2導(dǎo)數(shù)與應(yīng)用（1/2）

2023-08-26 18:41 作者:夏莉家的魯魯 0人讀過 | 我要投稿

0.2.1 導(dǎo)數(shù)的定義與表示

物理學(xué)中，我們經(jīng)常需要求一些物理量的變化率，而數(shù)學(xué)中的導(dǎo)函數(shù)是專門研究變化率的工具。

令 $y%3Df(x)$ 是關(guān)于 $x$ 的函數(shù)，則在任一點(diǎn)， $f(x)$ 關(guān)于 $x$ 的導(dǎo)函數(shù)的定義如下：

$%5Clim_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%5Cquad%20or%5Cquad%20%5Clim_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7Bf(x%2B%5CDelta%20x)-f(x)%7D%7B%5CDelta%20x%7D$

求導(dǎo)函數(shù)的過程簡稱求導(dǎo)，而 $f(x)$ 稱為導(dǎo)函數(shù)對(duì)應(yīng)的原函數(shù).

導(dǎo)函數(shù)的表示符號(hào)有很多，物理中常用的有2種：

$%5Cdfrac%7Bdf%7D%7Bdx%7D%2Cf'(x)$

由此我們可以得到

$v%3D%5Cdfrac%7Bdx%7D%7Bdt%7D%3Dx'(t)%2Ca%3D%5Cdfrac%7Bdv%7D%7Bdt%7D%3Dv'(t)$

特別地，在物理中，如果是對(duì)時(shí)間求導(dǎo)，我們可以在物理量上加一點(diǎn)表示：

$%5Cdot%20x%5Cequiv%20x'(t)%2C%5Cdot%20v%5Cequiv%20v'(t)$

不難看出，函數(shù)的導(dǎo)函數(shù)也可以有自己的導(dǎo)函數(shù)，我們可以把求導(dǎo)函數(shù)的次數(shù)叫做階，如果對(duì) $x$ 求2次導(dǎo)數(shù)，就叫做求 $x$ 的二階導(dǎo)數(shù)。對(duì)應(yīng)的符號(hào)分別表示為：

$%5Cdfrac%7Bd%5E2f%7D%7Bdx%5E2%7D%3D%20f''(x)$

如果是對(duì)時(shí)間求2次導(dǎo)數(shù)，可以在物理量上方加2個(gè)點(diǎn)： $%5Cddot%20x$

由基本定義以及極限的運(yùn)算，我們可以求得一些基本初等函數(shù)的導(dǎo)數(shù)：

$(x%5Ea)'%3Dax%5E%7Ba-1%7D$

當(dāng) $x%5Cne%200$ 時(shí)，

$(x%5Ea)'%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B(x%2B%5CDelta%20x)%5Ea-x%5Ea%7D%7B%5CDelta%20x%7D%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7Dx%5Ea%5Cdfrac%7B(1%2B%5Cfrac%7B%5CDelta%20x%7D%7Bx%7D)%5Ea-1%7D%7B%5CDelta%20x%7D$

$%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7Bx%5Eaa%5Cfrac%7B%5CDelta%20x%7D%7Bx%7D%7D%7B%5CDelta%20x%7D%3Dax%5E%7Ba-1%7D$

當(dāng) $x%3D0$ 時(shí)，結(jié)果符合上式.

$(a%5Ex)'%3Da%5Ex%5Cln%20a$

$(a%5Ex)'%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7Ba%5E%7Bx%2B%5CDelta%20x%7D-a%5Ex%7D%7B%5CDelta%20x%7D%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7Da%5Ex%5Cdfrac%7Ba%5E%7B%5CDelta%20x%7D-1%7D%7B%5CDelta%20x%7D$

$%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7Da%5Ex%5Cdfrac%7Be%5E%7B(%5Cln%20a)%5E%7B%5CDelta%20x%7D%7D-1%7D%7B%5CDelta%20x%7D%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7Da%5Ex%5Cdfrac%7B%5Cln%20a%5CDelta%20x%7D%7B%5CDelta%20x%7D%3Da%5Ex%5Cln%20a$

特殊地，當(dāng)a=e時(shí)，(e^x)'=e^x.

$(log_ax)'%3D%5Cdfrac%7B1%7D%7Bx%5Cln%20a%7D$

$(%5Clog_ax)'%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B%5Clog_a(x%2B%5CDelta%20x)-%5Clog_ax%7D%7B%5CDelta%20x%7D%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B1%7D%7B%5CDelta%20x%7D%7B%5Clog_a(%5Cdfrac%7Bx%2B%5CDelta%20x%7D%7Bx%7D)%7D$

$%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B1%7D%7B%5CDelta%20x%7D%7B%5Clog_a(1%2B%5Cdfrac%7B%5CDelta%20x%7D%7Bx%7D)%7D%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B1%7D%7Bx%7D%5Cdfrac%7Bx%7D%7B%5CDelta%20x%7D%5Clog_a(1%2B%5Cdfrac%7B%5CDelta%20x%7D%7Bx%7D)$

$%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B1%7D%7Bx%7D%5Clog_a(%5Cdfrac%7Bx%2B%5CDelta%20x%7D%7Bx%7D)%5E%5Cfrac%7B1%7D%7B%5CDelta%20x%7D%3D%5Cdfrac%7B1%7D%7Bx%7D%5Clog_ae%3D%5Cdfrac%7B1%7D%7Bx%5Cln%20a%7D$

特殊地，當(dāng) $a%3De$ 時(shí)，

$(%5Cln%20x)'%3D%5Cdfrac%7B1%7D%7Bx%7D$

$(%5Csin%20x)'%3D%5Ccos%20x$

$(%5Csin%20x)'%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B%5Csin(x%2B%5CDelta%20x)-%5Csin%20x%7D%7B%5CDelta%20x%7D$

$%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B%5Csin%20x%5Ccos(%5CDelta%20x)%2B%5Ccos%20x%5Csin(%5CDelta%20x)-%5Csin%20x%7D%7B%5CDelta%20x%7D$

$%3D%5Clim%5Climits_%7B%5CDelta%20x%5Crightarrow0%7D%5Cdfrac%7B%5Ccos%20x%5Csin(%5CDelta%20x)%7D%7B%5CDelta%20x%7D%3D%5Ccos%20x$

0.2.2 導(dǎo)數(shù)運(yùn)算法則

利用導(dǎo)函數(shù)的定義，我們?nèi)菀椎玫揭韵逻\(yùn)算法則：

四則運(yùn)算

1.加減

$%5Bf(x)%5Cpm%20g(x)%5D'%3Df'(x)%5Cpm%20g'(x)$

2.數(shù)乘

$%5B%5Calpha%20f(x)%5D'%3D%5Calpha%20f'(x)$

3.乘法

$%5Bf(x)%5Ccdot%20g(x)%5D'%3Df'(x)g(x)%2Bf(x)g'(x)$

4.除法

$%5B%5Cdfrac%7Bf(x)%7D%7Bg(x)%7D%5D'%3D%5Cdfrac%7Bf'(x)g(x)-f(x)g'(x)%7D%7Bg%5E2(x)%7D$

復(fù)合函數(shù)

$%5C%7Bf%5Bg(x)%5D%5C%7D'%3Df'(g)%5Ccdot%20g'(x)$

復(fù)合函數(shù)求導(dǎo)一直是難點(diǎn)中的難點(diǎn)，比如求 $y%3Dsin%5E2x$ 的導(dǎo)函數(shù)，需要注意 $y$ 是正弦函數(shù)和冪函數(shù)的復(fù)合函數(shù)，令 $u%3D%5Csin%20x$ ，則 $y%3Du%5E2$

$y'%5Bu(x)%5D%3Dy'(u)%5Ccdot%20u'(x)%3D2u%5Ccdot%5Ccos%20x%3D2%5Csin%20x%5Ccos%20x$

隱函數(shù)

形如 $y%3Df(x)$ 的函數(shù)，我們稱為顯函數(shù)，相應(yīng)的，有些函數(shù)并沒有表示成 $y%3Df(x)$ 的形式，比如 $xy%2Bx%2B1%3D0$ .我們稱之隱函數(shù)。

在對(duì)隱函數(shù)求導(dǎo)時(shí)，我們依然可以采用上述求導(dǎo)法則，比如上式，

$(xy%2Bx%2B1)'%3D0'$

$(y%2Bxy'%2B1%2B0)%3D0$

$y'%3D-%5Cdfrac%7By%2B1%7D%7Bx%7D$

代入 $y%3D-%5Cdfrac%7Bx%2B1%7D%7Bx%7D$ 得

$y'%3D%5Cdfrac%7B1%7D%7Bx%5E2%7D$

反函數(shù)

用因變量來表示自變量的函數(shù)，我們稱為對(duì)應(yīng)函數(shù)的反函數(shù)。 $y%3Df(x)$ 的反函數(shù)，可以表示為 $x%3Df%5E%7B-1%7D(y)%0A%0A$ 反函數(shù)導(dǎo)數(shù)等于原函數(shù)導(dǎo)數(shù)的倒數(shù)

$%5Cdfrac%7Bdy%7D%7Bdx%7D%3D%5Cdfrac%7B1%7D%7B%5Cdfrac%7Bdx%7D%7Bdy%7D%7D%5Cquad%20or%5Cquad%20f'(x)%3D%5Cdfrac%7B1%7D%7B%5Bf%5E%7B-1%7D(y)%5D'%7D$

比如對(duì)數(shù)函數(shù)的導(dǎo)數(shù)

$(%5Clog_ax)'%3D%5Cdfrac%7B1%7D%7B(a%5Ey)'%7D%3D%5Cdfrac%7B1%7D%7B(a%5Ey%5Cln%20a)%7D%3D%5Cdfrac%7B1%7D%7B(x%5Cln%20a%20)%7D$

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