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链式法则

链式法则告诉我们如何找到复合函数的导数。 了解关于复合函数的知识,并学习如何正确应用链式法则。
链式法则说:
start fraction, d, divided by, d, x, end fraction, open bracket, f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, close bracket, equals, f, prime, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, g, prime, left parenthesis, x, right parenthesis
它告诉我们如何对复合函数进行求导。

快速回顾什么是复合函数

一个函数能被写成f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis的形式则可被称为复合函数。换言之,它是一个函数内含另一个函数,或者说函数的函数。
比如说,start color #1fab54, cosine, left parenthesis, end color #1fab54, start color #e07d10, x, squared, end color #e07d10, start color #1fab54, right parenthesis, end color #1fab54是一个复合函数,因为如果start color #1fab54, f, left parenthesis, x, right parenthesis, equals, cosine, left parenthesis, x, right parenthesis, end color #1fab54start color #e07d10, g, left parenthesis, x, right parenthesis, equals, x, squared, end color #e07d10,则start color #1fab54, cosine, left parenthesis, end color #1fab54, start color #e07d10, x, squared, end color #e07d10, start color #1fab54, right parenthesis, end color #1fab54, equals, start color #1fab54, f, left parenthesis, end color #1fab54, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, start color #1fab54, right parenthesis, end color #1fab54
start color #e07d10, g, end color #e07d10start color #1fab54, f, end color #1fab54里的函数,所以我们称start color #e07d10, g, end color #e07d10为“内” 函数,start color #1fab54, f, end color #1fab54为“外”函数。
start color #1fab54, start underbrace, cosine, left parenthesis, space, start color #e07d10, start overbrace, x, squared, end overbrace, start superscript, start text, 内, end text, end superscript, space, end color #e07d10, right parenthesis, end underbrace, start subscript, start text, 外, end text, end subscript, end color #1fab54
另一方面,cosine, left parenthesis, x, right parenthesis, dot, x, squared不是一个复合函数。它是f, left parenthesis, x, right parenthesis, equals, cosine, left parenthesis, x, right parenthesisg, left parenthesis, x, right parenthesis, equals, x, squared乘积,但两个函数都不在对方的函数里。
问题1
g, left parenthesis, x, right parenthesis, equals, natural log, left parenthesis, sine, left parenthesis, x, right parenthesis, right parenthesis是一个复合函数吗?如果是,“内”函数和“外”函数分别是什么?
选出正确答案:

常见错误:无法识别出复合函数

通常,链式法则是唯一能对复合函数进行求导的方法。如果我们无法正确识别出必须要使用链式法则的复合函数,则我们无法正确求导。
另一方面,对不是复合函数的函数使用链式法则,也会得到错误的导数。
特别是超越函数(例如,三角函数,对数函数),学生通常会混淆natural log, left parenthesis, sine, left parenthesis, x, right parenthesis, right parenthesisnatural log, left parenthesis, x, right parenthesis, sine, left parenthesis, x, right parenthesis的乘积。
问题2
h, left parenthesis, x, right parenthesis, equals, cosine, squared, left parenthesis, x, right parenthesis是一个复合函数吗?如果是,“内”函数和“外”函数分别是什么?
选出正确答案:

想要更多练习?点击练习

常见错误:错误识别内外函数

即使学生能识别出复合函数,也有可能错误分解内外函数。这肯定会导致错误的求导。
比如说,在复合函数cosine, squared, left parenthesis, x, right parenthesis里,外函数是x, squared,内函数是cosine, left parenthesis, x, right parenthesis。学生通常会被这类函数弄糊涂,错误认为cosine, left parenthesis, x, right parenthesis是外函数。

链式法则使用实例

让我们来看一下如何用链式法则对h, left parenthesis, x, right parenthesis, equals, left parenthesis, 5, minus, 6, x, right parenthesis, start superscript, 5, end superscript进行求导。请注意h是一个复合函数:
h(x)=( 56x )5g(x)=56x内函数f(x)=x5外函数\begin{aligned} h(x) &= \greenD{\underbrace{(~\goldD{\overbrace{5-6x}^{\text{内}}~})^5}_{\text{外}}} \\\\ \goldD{g(x)}&=\goldD{5-6x} &&\text{内函数} \\\\ \greenD{f(x)}&=\greenD{x^5}&&\text{外函数} \end{aligned}
因为h是一个复合函数,我们可以用链式法则来求导:
start fraction, d, divided by, d, x, end fraction, open bracket, start color #1fab54, f, left parenthesis, end color #1fab54, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, start color #1fab54, right parenthesis, end color #1fab54, close bracket, equals, start color #11accd, f, prime, left parenthesis, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, right parenthesis, end color #11accd, dot, start color #ca337c, g, prime, left parenthesis, x, right parenthesis, end color #ca337c
用文字阐述出来,这个法则讲的是,复合函数的导数等于,内函数start color #e07d10, g, end color #e07d10代入外函数start color #11accd, f, prime, end color #11accd的值之导数,乘以内函数start color #ca337c, g, prime, end color #ca337c的导数。
在使用链式法则之前,我们需要先找到内外函数的导数:
g(x)=6f(x)=5x4\begin{aligned} \maroonD{g'(x)}&=\maroonD{-6} \\\\ \blueD{f'(x)}&=\blueD{5x^4} \end{aligned}
现在,让我们使用链式法则:
ddx[f(g(x))]=f(g(x))g(x)=5(56x)46=30(56x)4\begin{aligned} &\dfrac{d}{dx}\left[f\Bigl(g(x)\Bigr)\right] \\\\ =&\blueD{f'\Bigl(\goldD{g(x)}\Bigr)}\cdot\maroonD{g'(x)} \\\\ =&\blueD{5(\goldD{5-6x})^4} \cdot \maroonD{-6} \\\\ =&-30(5-6x)^4 \end{aligned}

练习使用链式法则

习题 3.A
  • 当前
第3习题集将一步步引导你完成对sine, left parenthesis, 2, x, cubed, minus, 4, x, right parenthesis的求导。
sine, left parenthesis, 2, x, cubed, minus, 4, x, right parenthesis的内外函数是什么?
选出正确答案:

问题 4
  • 当前
start fraction, d, divided by, d, x, end fraction, open bracket, square root of, cosine, left parenthesis, x, right parenthesis, end square root, close bracket, equals, question mark
选出正确答案:

想要更多练习?点击练习
问题5
  • 当前
xf, left parenthesis, x, right parenthesish, left parenthesis, x, right parenthesisf, prime, left parenthesis, x, right parenthesish, prime, left parenthesis, x, right parenthesis
minus, 19minus, 1minus, 5minus, 6
23minus, 116
G, left parenthesis, x, right parenthesis, equals, f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesis
G, prime, left parenthesis, 2, right parenthesis, equals
  • 你的答案是
  • 一个整数,例如 6
  • 一个最简真分数,如 3, slash, 5
  • 一个最简假分数,如 7, slash, 4
  • 一个混合带分数,例如 1, space, 3, slash, 4
  • 一个精确的十进位小数,例如0, point, 75
  • pi 的倍数, 例如 12\ \text{pi} 或 2/3\ \text{pi}$

想要更多练习?点击练习
问题6
小凯想求left parenthesis, 2, x, squared, minus, 4, right parenthesis, cubed的导数。以下是她的计算过程:
第1步:start color #1fab54, f, left parenthesis, x, right parenthesis, equals, x, cubed, end color #1fab54start color #e07d10, g, left parenthesis, x, right parenthesis, equals, 2, x, squared, minus, 4, end color #e07d10,那么start color #1fab54, left parenthesis, end color #1fab54, start color #e07d10, 2, x, squared, minus, 4, end color #e07d10, start color #1fab54, right parenthesis, cubed, end color #1fab54, equals, start color #1fab54, f, left parenthesis, end color #1fab54, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, start color #1fab54, right parenthesis, end color #1fab54
第2步: start color #11accd, f, prime, left parenthesis, x, right parenthesis, equals, 3, x, squared, end color #11accd
第3步:start color #11accd, f, prime, left parenthesis, end color #11accd, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, start color #11accd, right parenthesis, end color #11accd的导数是:
start fraction, d, divided by, d, x, end fraction, open bracket, start color #1fab54, left parenthesis, end color #1fab54, start color #e07d10, 2, x, squared, minus, 4, end color #e07d10, start color #1fab54, right parenthesis, cubed, end color #1fab54, close bracket, equals, start color #11accd, 3, left parenthesis, end color #11accd, start color #e07d10, 2, x, squared, minus, 4, end color #e07d10, start color #11accd, right parenthesis, squared, end color #11accd
小凯的计算过程正确吗?如果不对,她做错了哪里?
选出正确答案:

常见错误:忘记乘以内函数的导数

学生的一个常见错误是只对外函数进行求导,那就得到f, prime, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis,但正确的导数应为f, prime, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, g, prime, left parenthesis, x, right parenthesis

另一个常见错误:计算f, prime, left parenthesis, g, prime, left parenthesis, x, right parenthesis, right parenthesis

另一个常见错误是在求导f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis时,对两个函数同时求导,即f, prime, left parenthesis, g, prime, left parenthesis, x, right parenthesis, right parenthesis
这也是不正确的。在f, prime, left parenthesis, x, right parenthesis里的函数应为g, left parenthesis, x, right parenthesis,而不是g, prime, left parenthesis, x, right parenthesis
谨记: start color #1fab54, f, left parenthesis, end color #1fab54, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, start color #1fab54, right parenthesis, end color #1fab54的导数是start color #11accd, f, prime, left parenthesis, end color #11accd, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, start color #11accd, right parenthesis, end color #11accd, start color #ca337c, g, prime, left parenthesis, x, right parenthesis, end color #ca337c。不是start color #11accd, f, prime, left parenthesis, end color #11accd, start color #e07d10, g, left parenthesis, x, right parenthesis, end color #e07d10, start color #11accd, right parenthesis, end color #11accd,也不是start color #11accd, f, prime, left parenthesis, end color #11accd, start color #ca337c, g, prime, left parenthesis, x, right parenthesis, end color #ca337c, start color #11accd, right parenthesis, end color #11accd