数学高数篇其一——20220910

2022-09-11 17:50 作者:十一维的鱼 0人读过 | 我要投稿

说实话我有点纠结这个要不要归到高中篇里面，但再三思索里面涉及到的一些知识高中生可能没有学过，所以还是归到了高数篇

我的高中同学似乎也都已经开学了，只有我还在家里面发霉，只能写点东西打发时间，想给大家补充一下高中导数公式证明也算是补充了一个知识漏洞（据我所知高中阶段是不讲这个的，当然不排除某些强校会说）

为敲代码方便，导数我将用莱布尼茨表示法表示

y=x^a

$y%3Dx%5Ea%5C%5C%0A%5Cbegin%7Baligned%7D%5C%5C%5Cfrac%7Bdy%7D%7Bdx%7D%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7B(x%2B%5CDelta%20x)%5Ea-x%5Ea%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7B%5Csum%5Climits_%7Bn%3D0%7D%5EaC_a%5Enx%5E%7Ba-n%7D%5CDelta%20x%5En-x%5Ea%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7B%5Csum%5Climits_%7Bn%3D1%7D%5EaC_a%5Enx%5E%7Ba-n%7D%5CDelta%20x%5En%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Csum%5Climits_%7Bn%3D1%7D%5EaC_a%5Enx%5E%7Ba-n%7D%5CDelta%20x%5E%7Bn-1%7D%0A%5C%5C%26%3Dax%5E%7Ba-1%7D%0A%5Cend%7Baligned%7D$

y=a^x

$y%3Da%5Ex%5C%5C%0A%5Cbegin%7Baligned%7D%5C%5C%5Cfrac%7Bdy%7D%7Bdx%7D%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Ba%5E%7Bx%2B%5CDelta%20x%7D-a%5Ex%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D(%5Cfrac%7Ba%5E%7B%5CDelta%20x%7D-1%7D%7B%5CDelta%20x%7D)a%5Ex%0A%5C%5C%26%3Da%5Ex%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Bln(1%2Ba%5E%7B%5CDelta%20x%7D-1)%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3Da%5Exlna%0A%5Cend%7Baligned%7D$

y=sinx

$y%3Dsinx%5C%5C%0A%5Cbegin%7Baligned%7D%5C%5C%5Cfrac%7Bdy%7D%7Bdx%7D%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Bsin(x%2B%5CDelta%20x)-sinx%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Bsinxcos%5CDelta%20x%2Bcosxsin%5CDelta%20x-sinx%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7B(cos%5CDelta%20x-1)sinx%2B%5CDelta%20xcosx%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3Dcosx%0A%5Cend%7Baligned%7D%0A$

其余三角函数导数可利用诱导公式与导数运算推导

y=log_a^x $y%3Dlog_a%5Ex%5C%5C%20%5Cbegin%7Baligned%7D%5C%5C%5Cfrac%7Bdy%7D%7Bdx%7D%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Blog_a%5E%7B(x%2B%5CDelta%20x)%7D-log_a%5Ex%7D%7B%5CDelta%20x%7D%20%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7B%5Cfrac%7Bln%7B(x%2B%5CDelta%20x)%7D%7D%7Blna%7D-%5Cfrac%7Blnx%7D%7Blna%7D%7D%7B%5CDelta%20x%7D%20%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Bln%7B(1%2B%5Cfrac%7B%5CDelta%20x%7D%7Bx%7D)%7D%7D%7B%5CDelta%20xlna%7D%20%5C%5C%26%3D%5Cfrac%7B1%7D%7B%5CDelta%20xlna%7D%20%5Cend%7Baligned%7D$

导数运算法则推导

加法与减法从略

乘法

$%5Cbegin%7Baligned%7D%0A%5Ctext%5Bf(x)g(x)%5Ctext%5D%5E%7B'%7D%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Bf(x%2B%5CDelta%20x)g(x%2B%5CDelta%20x)-f(x)g(x)%7D%7B%5CDelta%20x%7D%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7Bf(x%2B%5CDelta%20x)g(x%2B%5CDelta%20x)-f(x)g(x%2B%5CDelta%20x)%2Bf(x)g(x%2B%5CDelta%20x)-f(x)g(x)%7D%7B%5CDelta%20x%7D%5C%5C%26%3Df(x)%5E%7B'%7Dg(x)%2Bg(x)%5E%7B'%7Df(x)%0A%5Cend%7Baligned%7D$

除法

$%5Cbegin%7Baligned%7D%5Ctext%5B%5Cfrac%7Bf(x)%7D%7Bg(x)%7D%5Ctext%5D%5E%7B'%7D%26%3D%5Clim_%7B%5CDelta%20x%5Cto0%7D%5Cfrac%7B%5Cfrac%7Bf(x%2B%5CDelta%20x)%7D%7Bg(x%2B%5CDelta%20x)%7D-%5Cfrac%7Bf(x)%7D%7Bg(x)%7D%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto%200%7D%5Cfrac%7B%5Cfrac%7Bf(x%2B%5CDelta%20x)%7D%7Bg(x%2B%5CDelta%20x)%7D-%5Cfrac%7Bf(x)%7D%7Bg(x%2B%5CDelta%20x)%7D-(%5Cfrac%7Bf(x)%7D%7Bg(x)%7D-%5Cfrac%7Bf(x)%7D%7Bg(x%2B%5CDelta%20x)%7D)%7D%7B%5CDelta%20x%7D%0A%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto%200%7D(%5Cfrac%7Bf(x)%5E%7B'%7D%7D%7Bg(x%2B%5CDelta%20x)%7D-%5Cfrac%7Bf(x)g(x)%5E%7B'%7D%7D%7Bg(x)g(x%2B%5CDelta%20x%EF%BC%89%7D)%5C%5C%26%3D%5Cfrac%7Bf(x)%5E%7B'%7Dg(x)-g(x)%5E%7B'%7Df(x)%7D%7Bg%5E2(x)%7D%0A%5Cend%7Baligned%7D%0A$

复合函数求导

$%5Cbegin%7Baligned%7D%5Ctext%5Bf(g(x))%5Ctext%5D%5E%7B'%7D%26%3D%5Clim_%7B%5CDelta%20x%5Cto%200%7D%5Cfrac%7Bf(g(x%2B%5CDelta%20x))-f(g(x))%7D%7B%5CDelta%20x%7D%5C%5C%26%3D%5Clim_%7B%5CDelta%20x%5Cto%200%7D%5Cfrac%7Bf(g(x%2B%5CDelta%20x))-f(g(x))%7D%7Bg(x%2B%5CDelta%20x)-g(x)%7D.%5Cfrac%7Bg(x%2B%5CDelta%20x)-g(x)%7D%7B%5CDelta%20x%7D%5C%5C%26%3Df%5E%7B'%7D(g(x))g(x)%0A%5Cend%7Baligned%7D$

根据莱布尼茨记法可以更简单的说明 $%5Cbegin%7Baligned%7Dy%26%3Df(u)%5C%5Cu%26%3Dg(x)%5C%5Cf(g(x))%5E%7B'%7D%3D%5Cfrac%7Bdy%7D%7Bdx%7D%26%3D%5Cfrac%7Bdy%7D%7Bdu%7D.%5Cfrac%7Bdu%7D%7Bdx%7D%3Df%5E%7B'%7D(x)g%5E%7B'%7D(x)%0A%5Cend%7Baligned%7D$

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