數(shù)學高數(shù)篇其一——20220910

2022-09-11 17:50 作者:十一維的魚 0人讀過 | 我要投稿

說實話我有點糾結(jié)這個要不要歸到高中篇里面，但再三思索里面涉及到的一些知識高中生可能沒有學過，所以還是歸到了高數(shù)篇

我的高中同學似乎也都已經(jīng)開學了，只有我還在家里面發(fā)霉，只能寫點東西打發(fā)時間，想給大家補充一下高中導(dǎo)數(shù)公式證明也算是補充了一個知識漏洞（據(jù)我所知高中階段是不講這個的，當然不排除某些強校會說）

為敲代碼方便，導(dǎo)數(shù)我將用萊布尼茨表示法表示

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

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

其余三角函數(shù)導(dǎo)數(shù)可利用誘導(dǎo)公式與導(dǎo)數(shù)運算推導(dǎo)

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

導(dǎo)數(shù)運算法則推導(dǎo)

加法與減法從略

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

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

復(fù)合函數(shù)求導(dǎo)

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

根據(jù)萊布尼茨記法可以更簡單的說明 $%5Cbegin%7Baligned%7Dy%26%3Df(u)%5C%5Cu%26%3Dg(x)%5C%5Cf(g(x))%5E%7B'%7D%3D%5Cfrac%7Bdy%7D%7Bdx%7D%26%3D%5Cfrac%7Bdy%7D%7Bdu%7D.%5Cfrac%7Bdu%7D%7Bdx%7D%3Df%5E%7B'%7D(x)g%5E%7B'%7D(x)%0A%5Cend%7Baligned%7D$

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數(shù)學高數(shù)篇其一——20220910

數(shù)學高數(shù)篇其一——20220910的評論 (共條)

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數(shù)學高數(shù)篇其一——20220910

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