8.9 经常衍生手段和权力规则

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  经常衍生物和权力规则

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  The power rule is a fantastic "shortcut" for finding the of basic polynomials. Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort - often in your head!
  ::权力规则是寻找基本多边协议的绝妙的“捷径 ” 。 在权力规则和常数衍生物的基本定义之间,大量多边协议衍生物可以毫不费力地被识别出来 — — 通常在你的脑中!

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  Constant Derivatives and the Power Rule
  ::经常衍生物和权力规则

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  In this lesson, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. Look for these theorems in boxes throughout the lesson.
  ::在此教训中, 我们将开发公式和定理, 以更高效、 更快捷的方式计算衍生物。 在整个课程中查找框中的这些定理 。

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  The Derivative of a Constant
  ::常数的衍生因素

  播放段落 Theorem: If $\begin{align*}f(x) = c\end{align*}$ where c is a constant, then $\begin{align*}f^{\prime}(x) = 0\end{align*}$ . ::定理 : 如果 f( x) =c 的 c 是常数, 那么 f_( x) =0 。 播放段落 Proof: $\begin{align*}f'(x)= \lim_{h \to 0}\frac {f(x+h)-f(x)}{h}=\lim_{h \to 0}\frac{c-c}{h} = 0\end{align*}$ . ::校对:f_(x)=limh_0f(x+h)-f(x)h=limh_0c-ch=0。

  Theorem: If $\begin{align*}c\end{align*}$ is a constant and $\begin{align*}f\end{align*}$ is differentiable at all $\begin{align*}x\end{align*}$ , then $\begin{align*}\frac {d}{dx}[cf(x)] = c\frac {d}{dx}[f(x)]\end{align*}$ . In simpler notation $\begin{align*}(cf)^{\prime} = c(f)^{\prime} = cf^{\prime}\end{align*}$

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  The Power Rule
  ::权力规则

  Theorem: (The Power Rule) If n is a positive integer, then for all real values of x $\begin{align*}\frac {d}{dx}[x^n] = nx^{n-1}\end{align*}$ .

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  Examples
  ::实例

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  Example 1
  ::例1

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  Find $\begin{align*}f^{\prime} (x)\end{align*}$ for $\begin{align*}f(x)=16\end{align*}$ .
  ::查找 f( x) =16 的 f_( x) 。

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  If $\begin{align*}f(x) = 16\end{align*}$ for all $\begin{align*}x\end{align*}$ , then $\begin{align*}f^{\prime} (x) = 0\end{align*}$ for all $\begin{align*}x\end{align*}$ .
  ::如果 f( x) =16 全部 x, 那么 f_( x) =0 全部 x 。

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  We can also write $\begin{align*}\frac{d}{dx}16 = 0\end{align*}$ .
  ::我们还可以写入 ddx16=0 。

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  Example 2
  ::例2

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  Find the derivative of $\begin{align*}f(x)=4x^3\end{align*}$ .
  ::查找 f( x) =4x3 的衍生物 。

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  $\begin{align*}\frac {d}{dx}\left [{4x^3} \right]\end{align*}$ ..... Restate the function
  ::ddx[4x3] ....

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  $\begin{align*}4 \frac{d}{dx}\left [{x^3} \right]\end{align*}$ ..... Apply the commutative law
  ::4ddx[x3] ....应用通货法

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  $\begin{align*}4 \left [{3x^2} \right]\end{align*}$ ..... Apply the power Rule
  ::4[3x2].应用权力规则

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  $\begin{align*}12x^2\end{align*}$ ..... Simplify
  ::12x2.... 简化

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  Example 3
  ::例3

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

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  $\begin{align*}\frac {d}{dx} \left [\frac{-2}{x^4} \right]\end{align*}$ ..... Restate
  ::ddx[- 2x4] .... 重报

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  $\begin{align*}\frac {d}{dx}\left [{-2x^{-4}} \right]\end{align*}$ ..... Rules of exponents
  ::ddx[-2x-4]. 引言人规则

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  $\begin{align*}-2 \frac {d}{dx}\left [{x^{-4}} \right]\end{align*}$ ..... By the commutative law
  ::-2dx[x-4].根据通货法

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  $\begin{align*}-2 \left [{-4x^{-4-1}} \right]\end{align*}$ ..... Apply the power rule
  ::--2[-4x-4-1] ...。应用权力规则

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  $\begin{align*}-2 \left [{-4x^{-5}} \right]\end{align*}$ ..... Simplify
  ::--2[-4x-5]. 简化

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  $\begin{align*}8x^{-5}\end{align*}$ ..... Simplify again
  ::8x-5. 再次简化

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  $\begin{align*}\frac {8}{x^5}\end{align*}$ ..... Use rules of exponents
  ::8x5. 使用指数者的规则

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  Example 4
  ::例4

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  Find the derivative of  $\begin{align*}f(x)=x\end{align*}$ .
  ::查找 f( x) =x 的衍生物 。

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  Special application of the power rule:
  ::权力规则的特殊适用:

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  $\begin{align*}\frac {d}{dx}[x] = 1x^{1-1} = x^0 = 1\end{align*}$
  ::ddx[x]=1x1 -1=x0=1

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  Example 5
  ::例5

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  Find the derivative of  $\begin{align*}f(x)=\sqrt{x}\end{align*}$ .
  ::查找 f( x) =x 的衍生物 。

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  Restate the function: $\begin{align*}\frac {d}{dx}[\sqrt{x}]\end{align*}$
  ::复述函数: ddx[x]

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  Using rules of exponents (from algebra): $\begin{align*}\frac {d}{dx}[x^{1/2}]\end{align*}$
  ::使用前言规则( 取自代数) : ddx [x1/ 2]

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  Apply the power rule: $\begin{align*}\frac {1}{2}x^{1/2-1}\end{align*}$
  ::应用权力规则: 12x1/2- 1

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  Simplify: $\begin{align*}\frac {1}{2}x^{-1/2}\end{align*}$
  ::简化: 12x-1/2

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  Rules of exponents: $\begin{align*}\frac{1}{2x^{1/2}}\end{align*}$
  ::规则:12x1/2

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  Simplify: $\begin{align*}\frac {1}{2\sqrt{x}}\end{align*}$
  ::简化: 12x

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  Example 6
  ::例6

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  Find the derivative of  $\begin{align*}f(x)=\frac{1}{x^{3}}\end{align*}$ .
  ::查找 f( x) = 1x3 的衍生物 。

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  Restate the function: $\begin{align*}\frac {d}{dx}\left [ \frac{1}{x^3} \right ]\end{align*}$
  ::复述函数: ddx[1x3]

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  Rules of exponents: $\begin{align*}\frac {d}{dx}\left [{x^{-3}} \right ]\end{align*}$
  ::引言人规则: ddx[x-3]

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  Power rule: $\begin{align*}-3x^{-3-1}\end{align*}$
  ::权力规则:-3x-3-3-1

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  Simplify: $\begin{align*}-3x^{-4}\end{align*}$
  ::简化:- 3x- 4

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  Rules of exponents: $\begin{align*}\frac {-3}{x^4}\end{align*}$
  ::引言人规则:-3x4

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  Review
  ::回顾

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  1. State the power rule.
     ::国家的权力统治。

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  Find the derivative:
  ::查找衍生物 :

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  2. $\begin{align*}y = 5x^7\end{align*}$
     ::y=5x7 y=5x7
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  3. $\begin{align*}y = -3x\end{align*}$
     ::y 3x
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  4. $\begin{align*}f(x) = \frac{1} {3} x + \frac{4} {3}\end{align*}$
     :xx)=13x+43
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  5. $\begin{align*}y = x^4 - 2x^3 - 5\sqrt{x} + 10\end{align*}$
     ::y=x4 - 2x3 - 5x+10 y=x4 - 2x3 - 5x+10
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  6. $\begin{align*}y = (5x^2 - 3)^2\end{align*}$
     ::y=( 5x2- 3) 2
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  7. Given $\begin{align*}y(x)= x^{-4\pi^2}\end{align*}$ , find the derivative when $\begin{align*} x = 1\end{align*}$ .
     ::give y(x) =x- 42, 在 x=1 时找到衍生物 。
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  8. $\begin{align*}y(x) = 5\end{align*}$
     ::y(x)=5
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  9. Given $\begin{align*}u(x)= x^{-5\pi^3}\end{align*}$ , what is $\begin{align*} u'(2)\end{align*}$ ?
     ::鉴于 u(x) =x-53,什么是 u(2)?
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  10. Given  $\begin{align*}d(x)= x^{-0.37}\end{align*}$ , what is $\begin{align*} d'(1)\end{align*}$ ?
      ::鉴于d(x)=x-0.37,什么是d(1)?
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  11. $\begin{align*} g(x) = x^{-3}\end{align*}$
      ::g(x)=x-3
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  12. $\begin{align*}u(x) = x^{0.096}\end{align*}$
      ::u(x)=x0.096
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  13. $\begin{align*}k(x) = x-0.49\end{align*}$
      :xx)=x-0.49
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  14. $\begin{align*} y = x^{-5\pi^3}\end{align*}$
      ::y=x - 5=3

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。