WEBVTT

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有人要求讲一下

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证明√x的导数

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所以这集快速证明一下

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√x的导数

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由导数的定义可知

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函数√x的导数

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换颜色 更多样一些

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它等于Δx趋于0时的极限

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有人也说h趋于0

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或d趋于0 这里用Δx

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Δx趋于0时

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然后f(x+Δx)

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此处这是f(x)

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所以它是√(x+Δx)-f(x)

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此处即√x

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整体/x的变化量Δx

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现在看这个式子

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还不能化简

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使它变得适于分析

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那么分子和分母同乘以

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分子的共轭根式 什么意思呢？

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把它写下来

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Δx趋于0时的极限

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把这再写下来

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即√(x+Δx)

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-√x

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整体/Δx

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换颜色 将它乘以

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√(x+Δx)

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+√x 除以

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√(x+Δx)+√x

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这只是1 显然可以相乘

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假设x和Δx都不为0

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这是一个定值 为1

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可以相乘

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这是1/1 将它乘以这个式子

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得到当Δx趋于0时

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这是(a-b)(a+b)

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在这旁边列出来

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(a+b)(a-b)

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=a^2-b^2

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这是(a+b)(a-b)

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所以它等于a^2

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那么这个量的平方是什么

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两个都可以 都代表a

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是x+Δx

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那么得到x+Δx

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然后b^2呢？

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-√x类比为b

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那么(√x)^2=x

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整体/(Δx・

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(√(x+Δx)+√x))

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看看怎么化简

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这里有x和-x

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两者抵消了 x-x

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然后分子和分母上剩下的

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都有Δx

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分子和分母同除以Δx

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都变为1

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所以它= 写的小点

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因为没地方了

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=Δx趋于0时的极限 1/

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显然只有假设Δx--

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一开始除以Δx

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所以它不为0 只是趋于0

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那么得到√(x+Δx)

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+√x

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现在直接求Δx趋于0时

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它的极限

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只需设Δx=0

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即趋于0的含义

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那么等于1/√x

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对吗？Δx=0 可以忽略

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取其极限一直到0

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然后这显然还是√x

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+√x

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=1/(2√x)

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=(1/2)x^(-1/2)

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刚才证明了x^(1/2)

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的导数为(1/2)x^(-1/2)

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它跟一般情况一致

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即求导

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x^n的导数为

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n・x^(n-1) 即使此处n=1/2

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也是合乎情况的

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未就所有的分数都给出证明 但这是开始

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这是常见的一个 √x

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好在证明起来不很复杂

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下一集再见