3.4 区别规则:常数和变数力

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• 选择节差别规则:常数和变数力

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  差别规则:常数和变数力

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  What is the slope of a horizontal line? What is the slope of any line that can be written in slope-intercept form? Can you relate these with a derivative? If you know the answers to these questions, then you already prepared to state a few differentiation rules. Think about your answers before you go on with this lesson. The rules presented here offer “shortcuts” to the limit definition for finding the of basic polynomials.
  ::水平线的斜坡是什么? 以斜坡截取形式写出的任何线的斜坡是什么? 您能否将这些斜坡与衍生物联系起来? 如果您知道这些问题的答案, 那么您已经准备说明几个区别规则 。 在您继续学习这个课程之前, 请先考虑一下答案 。 这里提出的规则为找到基本多元性提供了“ 捷径 ” 。

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  Differentiation Rules
  ::区别规则

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  In the following discussion, we will present some examples that provide the basis for formulas and theorems that will replace the use of the limit definition of the derivative with more efficient and quicker ways of finding the derivative. 
  ::在接下来的讨论中,我们将提出一些例子,为公式和理论提供基础,以更有效更快的方式找到衍生物,取代对衍生物的有限定义的使用。

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  Derivative of a Constant
  ::常量的衍生物

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  Let's first find the derivative of a constant function, $\begin{array}{r}f(x)=16\end{array}$ , using the definition of the derivative.
  ::首先让我们使用衍生物的定义 来找到一个恒定函数的衍生物 f(x)=16, f(x)=16。

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  $$\begin{array}{rl}\frac{dy}{dx}& =\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}\\ \frac{dy}{dx}& =\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}\\ & =\underset{h\to 0}{lim}\frac{16-16}{h}=\underset{h\to 0}{lim}\frac{0}{h}\\ & =0\end{array}$$

  
  ::uddx=limh_0f(x+h)-f(x)hdydx=limh_0f(x+h)-f(x)h=limh_016-16h=limh_00h=0

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  The above limit formulation can be generalized to the following derivative rule:
  ::上述限额规定可概括为以下衍生规则:

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  If $\begin{array}{r}f(x)=c\end{array}$  where  $\begin{array}{r}c\end{array}$ is a constant, then $\begin{array}{r}{f}^{\mathrm{\prime }}(x)=0\end{array}$ .
  ::如果f(x)=c,其中c为常数,则f_(x)=0。

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  Derivative of the Product of a Constant and a Function
  ::常数和函数产品的衍生物

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  Find the derivative of the function $\begin{array}{r}g(x)=cf(x)\end{array}$ , where  $\begin{array}{r}c\end{array}$ is a constant and  $\begin{array}{r}f(x)\end{array}$ is a function that has a derivative for all  $\begin{array}{r}x\end{array}$ of interest.
  ::查找函数 g(x) =cf(x) 的衍生物,其中 c 是常数, f(x) 是一个函数, 包含所有 x 的衍生物 。

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  Using the definition of the derivative as a limit:
  ::使用衍生品的定义作为限额:

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  $$\begin{array}{rl}\frac{dy}{dx}& =\underset{h\to 0}{lim}\frac{g(x+h)-g(x)}{h}\\ & =\underset{h\to 0}{lim}\frac{cf(x+h)-cf(x)}{h}\\ & =\underset{h\to 0}{lim}c\cdot \left(\frac{f(x+h)-f(x)}{h}\right)=\underset{h\to 0}{lim}c\cdot \underset{h\to 0}{lim}\left(\frac{f(x+h)-f(x)}{h}\right)\\ & =c{f}^{\mathrm{\prime }}(x)\end{array}$$

  
  ::uddx=limh=0g(x+h)-g(x)-g(x)h=limh=0cf(x+h)-c(f(x)h)-f(x)h)-f(x)-f(x)h)=limh=0climh*0(f(x+h)-f(x)h)=cf=x

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  The above limit formulation can be generalized to the following derivative rule:
  ::上述限额规定可概括为以下衍生规则:

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  If $\begin{array}{r}c\end{array}$  is a constant and $\begin{array}{r}f\end{array}$  is differentiable at all $\begin{array}{r}x\end{array}$ , then $\begin{array}{r}\frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]\end{array}$ .
  ::如果 c 是一个常数, f 完全可以区分 x, 那么 ddx [(x) = cddx [f(x)] 。

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  In simpler notation $\begin{array}{r}(cf{)}^{\mathrm{\prime }}=c(f{)}^{\mathrm{\prime }}=c{f}^{\mathrm{\prime }}\end{array}$
  ::更简单的注解(参考)c(f)___________________________________________________________________________________________________________________________________

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  The Power Rule
  ::权力规则

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  Find the derivative of  $\begin{array}{r}f(x)=x^n\end{array}$ where  $\begin{array}{r}n\end{array}$ is a positive integer.
  ::查找 f( x) =xn 的衍生物, n 是正整数 。

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  $$\begin{array}{rl}\frac{dy}{dx}& =\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}\\ & =\underset{h\to 0}{lim}\frac{(x+h{)}^n-x^n}{h}\dots \text{Use the Binomial Theorem to expand}(x+h{)}^n.\\ & =\underset{h\to 0}{lim}\frac{\left(x^n+n{x}^{n-1}h+\frac{n(n-1)}{2!}{x}^{n-2}{h}^2+\frac{n(n-1)(n-2)}{3!}{x}^{n-3}{h}^3+\cdots nx{h}^{n-1}+h^n\right)-x^n}{h}\\ & =\underset{h\to 0}{lim}\frac{n{x}^{n-1}h+\frac{n(n-1)}{2!}{x}^{n-2}{h}^2+\frac{n(n-1)(n-2)}{3!}{x}^{n-3}{h}^3+\cdots nx{h}^{n-1}+h^n}{h}\\ & =\underset{h\to 0}{lim}\left(n{x}^{n-1}+\frac{n(n-1)}{2!}{x}^{n-2}h+\frac{n(n-1)(n-2)}{3!}{x}^{n-3}{h}^2+\cdots nx{h}^{n-2}+h^{n-1}\right)\\ & =n{x}^{n-1}\end{array}$$

  
  :x+h)-(x)h=limh0(x+h)-f(x)h=limh0(x+h)0(x+h)n-xxxxxxxxxxxxxxx=limh0(x+h)0(x+xxx0(xxxxxxxxxxxxxxxxxxxxxxxxx+(x+)0(x+)xxxxxxx=xxxxxxxxx=0(n-1)nxxxx0(xxxxxxxxxxxxxx00(xxxxxxxxxxx+N-1)xxxx0(xxxxxxxxx-1+n(n-1)xxxxx-2h(n-1)(n-2xx-2h)xxxxx-2(n-2xx-2-h-h)1(n-1)xxxxxx

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  The above limit formulation can be generalized to the following derivative rule that is called the  Power Rule :
  ::上述限额规定可概括为以下称为权力规则的衍生规则:

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  If $\begin{array}{r}n\end{array}$ is a real number, then for all real values of $\begin{array}{r}x\end{array}$
  ::如果 n 是真实数字, 那么对于所有 x 的真实值

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  $$\begin{array}{r}\frac{d}{dx}[x^n]=n{x}^{n-1}.\end{array}$$

  
  ::dx[xn]=nxn-1。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to think about shortcuts in taking derivatives. Can you see why the Constant and Power Rules are so handy and fairly easy to remember? There is a nice order to the application to the rules as follows:
  ::早些时候,有人要求你考虑获取衍生物的捷径。 你可以看到《常数和权力规则》为什么如此简便和容易记住吗?对规则的适用有一个良好的顺序,如下所示:

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  - The derivative of a constant function(horizontal line, $\begin{array}{r}y=a\end{array}$ )  is $\begin{array}{r}{y}^{\mathrm{\prime }}=0\end{array}$ .
  ::- 恒定函数(横向线,y=a)的衍生物为y0。

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  - The derivative of a linear function  $\begin{array}{r}(y=ax+b)\end{array}$ is a constant, $\begin{array}{r}{y}^{\mathrm{\prime }}=a\end{array}$ .
  ::- 线性函数(y=ax+b)的衍生物是一个常数,ya。

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  - The derivative of a quadratic function  $\begin{array}{r}(y=a{x}^2)\end{array}$ is a linear function, $\begin{array}{r}{y}^{\mathrm{\prime }}=2ax\end{array}$
  ::- 二次函数(y=ax2)的衍生物是一个线性函数,y2ax

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  - The derivative of a cubic function  $\begin{array}{r}(y=a{x}^3)\end{array}$ is a quadratic function, $\begin{array}{r}{y}^{\mathrm{\prime }}=3a{x}^2\end{array}$ .

  $$\begin{array}{r}\cdot \\ \cdot \\ \cdot \end{array}$$

  
  ::- 立方函数(y=ax3)的衍生物是一个二次函数,y3ax2.

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   Example 2
  ::例2

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  Find the derivative of   $\begin{array}{r}f(x)=\frac{-2}{x^4}\end{array}$
  ::查找 f(x)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ f\\\\\\\\\\\\\\\\ f\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  $$\begin{array}{rlrl}& \frac{d}{dx}\left[\frac{-2}{x^4}\right]& & \dots \text{Restate}\\ & \frac{d}{dx}[-2x^{-4}]& & \dots \text{Rules of exponents}\\ & -2\frac{d}{dx}[x^{-4}]& & \dots \text{Derivative Rule: Product of a constant and function}\\ & -2[-4x^{-4-1}]& & \dots \text{Derivative Rule: Power Rule}\\ & 8x^{-5}& & \dots \text{Simplify}\\ & \frac{8}{x^5}& & \dots \text{Rules of exponents}\end{array}$$

  
  ::ddx[-2x4].Restaceddx[-2x-4].Expents -2ddx[x-4].derivation rules: a certain and follows -2[-4x-4-4-1].derivation rules: power rules8x-5... 简化8x5... expents rules

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  Example 3
  ::例3

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  Find the derivative of   $\begin{array}{r}f(x)=x^3\end{array}$
  ::查找 f( x) =x3 的衍生物

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  By the power rule: If $\begin{array}{r}f(x)=x^3\end{array}$ then $\begin{array}{r}{f}^{\mathrm{\prime }}(x)=(3)x^{3-1}=3x^2\end{array}$
  ::根据权力规则: Iff(x) =x3thenf}(x) = (3)x3- 1=3x2

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  Example 4
  ::例4

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  Find the derivative of   $\begin{array}{r}f(x)=x\end{array}$
  ::查找 f( x) =x 的衍生物

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  By the power rule: $\begin{array}{r}\frac{d}{dx}[x]=1x^{1-1}=x^0=1\end{array}$
  ::根据权力规则:ddx[x]=1x1 -1=1=x0=1

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  Example 5
  ::例5

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  Find the derivative of   $\begin{array}{r}f(x)=\sqrt{x}\end{array}$
  ::查找 f( x) =x 的衍生物

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  Restate the function: $\begin{array}{r}\frac{d}{dx}\left[\sqrt{x}\right]=\end{array}$
  ::恢复函数: ddx[x]=

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  Using rules of exponents: $\begin{array}{r}\frac{d}{dx}[x^{\frac{1}{2}}]\end{array}$
  ::使用引言规则: ddx[x12]

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  Apply the Power Rule: $\begin{array}{r}\frac{1}{2}{x}^{\frac{1}{2}-1}\end{array}$
  ::适用权力规则:12x12-1

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  Simplify: $\begin{array}{r}\frac{1}{2}{x}^{-\frac{1}{2}}\end{array}$
  ::简化: 12x- 12

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  Rules of exponents: $\begin{array}{r}\frac{1}{2x^{\frac{1}{2}}}\end{array}$
  ::代表规则:12x12

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  Simplify: $\begin{array}{r}\frac{1}{2\sqrt{x}}\end{array}$
  ::简化: 12x

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  Review
  ::回顾

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  1. State the Power Rule.
     ::国家权力统治。

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  For #2-15, find the derivative. 
  ::2 -15,找到衍生物

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  2. $\begin{array}{r}y=5x^7\end{array}$
     ::y=5x7 y=5x7
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  3. $\begin{array}{r}y=-3x\end{array}$
     ::y 3x
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  4. $\begin{array}{r}f(x)=\frac{1}{3}x+\frac{4}{3}\end{array}$
     :xx)=13x+43
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  5. $\begin{array}{r}y=x^4-2x^3-5\sqrt{x}+10\end{array}$
     ::y=x4 - 2x3 - 5x+10 y=x4 - 2x3 - 5x+10
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  6. $\begin{array}{r}y=(5x^2-3{)}^2\end{array}$
     ::y=( 5x2- 3) 2
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  7. Given  $\begin{array}{r}y(x)=x^{-4{\pi }^2}\end{array}$ when $\begin{array}{r}x=1\end{array}$
     ::x=1 时给定 y( x) =x- 42
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  8. $\begin{array}{r}y(x)=5\end{array}$
     ::y(x)=5
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  9. Given  $\begin{array}{r}u(x)=x^{-5{\pi }^3}\end{array}$   when  $\begin{array}{r}x=2\end{array}$  
     ::x=2 时给定 u( x) =x- 53
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  10. $\begin{array}{r}y=\frac{1}{5}\end{array}$  when $\begin{array}{r}x=4\end{array}$
      ::y=15 x=4 时 y=15
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  11. Given  $\begin{array}{r}d(x)=x^{-0.37}\end{array}$ when  $\begin{array}{r}x=1\end{array}$
      ::给定 d( x) =x- 0.37 当 x=1 时
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  12. $\begin{array}{r}g(x)=x^{-3}\end{array}$
      ::g(x)=x-3
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  13. $\begin{array}{r}u(x)=x^{0.096}\end{array}$
      ::u(x)=x0.096
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  14. $\begin{array}{r}k(x)=x-0.49\end{array}$
      :xx)=x-0.49
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  15. $\begin{array}{r}y=x^{-5{\pi }^3}\end{array}$
      ::y=x - 5=3

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  Review (Answers)
  ::回顾(答复)

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