3.9 权力职能的衍生因素

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• 选择节指数函数的衍生因素

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  指数函数的衍生因素

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  Exponential functions, and the rate of change, are used to model many real-world situations such as population growth, radioactive half-life decay, attenuation of electromagnetic signals in media, and financial transactions. Do you know how to write general exponential equations for the growth of a population that doubles every 5 years, and its rate of change? 
  ::指数功能和变化率被用来模拟许多现实世界局势,如人口增长、放射性半衰期衰减、媒体电磁信号减弱以及金融交易。 你知道如何为每五年翻一番的人口增长率及其变化速度写出一般指数方程式吗?

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  Derivatives of Exponential Functions
  ::指数函数的衍生因素

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  An exponential function $\begin{align*}f(x) \end{align*}$ has the form:
  ::指数函数 f( x) 具有窗体 :

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  $$\begin{align*}f(x) = {b^x},\end{align*}$$
  :xx=bx,

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  where  $\begin{align*}b\end{align*}$ is called the base and is a positive, real number.
  ::b 称为基数,是正数,实际数。

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  The figure below shows a few exponential function graphs for $\begin{align*}0 < b \le 10\end{align*}$ . It is very clear that the sign of the derivative of an exponential depends on the value of  $\begin{align*}b\end{align*}$ . If $\begin{align*}0 < b < 1\end{align*}$ , the value of the derivative of the function (slope of the tangent line) will be negative because the function is always decreasing as  $\begin{align*}x\end{align*}$ increases. For $\begin{align*}b > 1\end{align*}$ , the derivative of the function will always be positive because the function increases as  $\begin{align*}x\end{align*}$ increases.
  ::下图显示 0<b10 的几张指数函数图形。非常清楚,指数衍生物的符号取决于 b 值。如果 0<b<1,函数衍生物(正切线的斜面)的值为负值,因为函数随着 x 的增加总是在下降。对于 b>1 ,函数的衍生物总是正值,因为函数随着 x 的增加而增加。

  

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  But, what is the derivative of an exponential function? We can take the following steps to find an expression for $\begin{align*}\frac{d}{{dx}}[{b^x}]\end{align*}$ by using the definition of the derivative :
  ::但是,指数函数的衍生物是什么?我们可以采取以下步骤,通过使用衍生物的定义来找到 ddx [bx] 的表达方式:

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  $$\begin{align*}\frac{d}{dx}[b^x] &=\lim_{h \to 0}\frac{b^{x+h}-b^x}{h} && .... \text{Limit definition of the derivative} \\ &=\lim_{h \to 0}\frac{b^xb^h-b^x}{h} && .... \text{Exponent property} \\ &=\lim_{h \to 0}\frac{b^h-1}{h} \cdot b^x && .... \text{Factoring} \\ &=\left(\lim_{h \to 0} \frac{b^h-1}{h} \right) \cdot b^x && .... \text{Limit of a product property}\end{align*}$$
  ::ddx[ bx] =limh0bx+h- bxh.... 衍生物=limh0bxbh- bxh- bxh.... 显性属性=limh0bh- 1hx.... 引力= (limh%0bh- 1h) bx. 产品属性的显性

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  The result above shows $\begin{align*}\frac{d}{{dx}}[{b^x}]\end{align*}$ depends on the product of $\begin{align*}\left(\lim_{h \to 0} \frac{b^h-1}{h} \right)\end{align*}$ and the original function. But what is $\begin{align*}\left(\lim_{h \to 0} \frac{b^h-1}{h} \right)\end{align*}$ ? There are a number of ways to evaluate this limit, but for now let’s take a quick look at the behavior of $\begin{align*}\left(\frac{b^h-1}{h}\right)\end{align*}$ . This function is graphed below for a number of values of  $\begin{align*}b\end{align*}$ , and the limit at  $\begin{align*}h=0\end{align*}$ is indicated by the points $\begin{align*}A-E\end{align*}$ .
  ::以上结果显示 ddx [bx] 取决于(limh0bh- 1h) 的产物和原始函数 。 但是, 什么是 (limh0bh- 1h) ? 有多种方法来评估这一限值, 但是现在让我们快速查看( bh- 1h) 的行为。 这个函数按b 的数值绘制图示如下, 点 A- E 表示 h=0 的限值 。

  

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  While it is not at all obvious: $\begin{align*}\lim_{h \to 0}\frac{b^h-1}{h}=\ln(b)\end{align*}$ . Remember the natural logarithm function $\begin{align*}y=\ln x\end{align*}$ on your calculator? The natural logarithm is the general logarithm function with base $\begin{align*}b=e=2.71828...\end{align*}$
  ::虽然并非显而易见 : limh0bh- 1h=ln(b) 。 记得计算器上的自然对数函数 y=lnx吗? 自然对数是 b= e= 2. 71828 的普通对数函数...

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  Given the exponential function $\begin{align*}f(x)=b^x\end{align*}$ , where the base  $\begin{align*}b\end{align*}$ is a positive, real number, then the general representation of the derivative of an exponential function is:
  ::考虑到指数函数 f(x) = bx, 基数 b 是正数, 实际数, 那么指数函数的衍生物的一般表示值是 :

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  $$\begin{align*}\frac{d}{dx}[b^x]=\ln b \cdot b^x\end{align*}$$
  ::ddx[bx]=lnbx

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  Adding the to the definition, given the exponential function $\begin{align*}f(x)=b^u\end{align*}$ , where $\begin{align*}u=g(x)\end{align*}$ and $\begin{align*}g(x)\end{align*}$ is a differentiable function, then:
  ::添加到定义中, 考虑到指数函数 f( x) =bu, 即 u=g( x) 和 g( x) 是一个可区别的函数, 那么 :

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  $$\begin{align*}\frac{d}{dx}[b^u]=(\ln b \cdot b^u)\frac{du}{dx}\end{align*}$$
  ::ddx[bu]=( lnbbu) dudx

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked what the general exponential equation for the growth of a population that doubles every 5 years is. 
  ::早些时候,有人问你,每五年翻一番的人口增长率的总指数方程式是多少。

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  A population $\begin{align*}P(t)\end{align*}$ that doubles every 5 years could be modeled as $\begin{align*}P(t)=P_0 2^{\frac{t}{5}}\end{align*}$ , where the variable $\begin{align*}t\end{align*}$ represents number of years since the population was at a level of $\begin{align*}P_0\end{align*}$ . Were you able to determine that the rate of change of $\begin{align*}P(t)\end{align*}$ is $\begin{align*}P^{\prime}(t)=\frac{P_0 \ln 2}{5} \cdot 2^{\frac{t}{5}}\end{align*}$ ?
  ::人口P(t)每5年翻一番的P(t)=P02t5,其中变量t表示自人口处于P0水平以来的年数。 您能否确定P(t)的变动率是P(t)=P0ln25}2t5?

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  Example 2
  ::例2

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  Given $\begin{align*}y=500 \cdot 0.7^x\end{align*}$ , what is $\begin{align*}\frac{dy}{dx}\end{align*}$ ?
  ::鉴于y=500 =0.7x,什么是?

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  $$\begin{align*}\frac{dy}{dx} &=\frac{d}{dx}[500 \cdot 0.7^x] \\ &=500 \frac{d}{dx}[0.7^x] \\ &=500[\ln(0.7) \cdot 0.7^x] \qquad .... \text{Use your calculator to find } \ln(0.7) \\ &=-178.3 \cdot 0.7^x \end{align*}$$
  ::dydx=ddx[500=0.7x] =500ddx[0.7x] =500[ln( 0.7) =0.7x]....使用您的计算器查找 In( 0.7) {178.3=0.7x

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  Hence, $\begin{align*}\frac{dy}{dx}=- 178.3 \cdot 0.7^x\end{align*}$ , and as expected, the slopes of all tangent lines are negative.
  ::因此,Didx178.30.7x,并如预期的那样,所有正切线的斜坡都是负的。

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  There is an important special case that you must know about
  ::有个很重要的特殊情况,你必须知道

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  Example 3
  ::例3

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  Given $\begin{align*}y = 500 \cdot e^x \end{align*}$ , what is $\begin{align*}\frac{dy}{dx}\end{align*}$ ?
  ::鉴于y=500 ex,什么是Didx?

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  $$\begin{align*}\frac{dy}{dx} &=\frac{d}{dx}[500 \cdot e^x] \\ &=500 \frac{d}{dx}[e^x] \\ &=500[\ln(e) \cdot e^x] \qquad .... \text{Use your calculator to find } \ln(e)\\ \frac{dy}{dx} &=\frac{d}{dx}[500 \cdot e^x] \\ &=500 [1 \cdot e^x] \\ &=500 \cdot e^x \end{align*}$$
  ::didx=ddx[ 500\ex] = 500ddx[ex] = 500 [ln(e)\\ex]....使用您的计算器查找 In(e) didx=ddx[500\ex] =500[1\ex] = 500xx

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  Hence, $\begin{align*}\frac{dy}{dx}=500 \cdot e^x\end{align*}$ , and this is just the original function. This exponential function, with base  $\begin{align*}e\end{align*}$ , is special: the rate of change (or slope of the tanget line) at any point is equal to the value of the function at that point.
  ::因此, dydx=500ex, 这只是最初的函数。 此指数函数, 以 base e 为单位, 具有特殊性: 任何一点的变化率( 或制革线的斜度) 等于该函数在那个点的值 。

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  Example 4
  ::例4

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  Given $\begin{align*}y=10 \cdot 2.5^{-3x^2}\end{align*}$ , what is $\begin{align*}\frac{dy}{dx}\end{align*}$ ?
  ::鉴于y=102.5-3x2,什么是?

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  $$\begin{align*}\frac{dy}{dx} &=\frac{d}{dx}[10 \cdot 2.5^{-3x^2}] \\ &=10 \frac{d}{dx}[2.5^{-3x^2}] \\ &=10 \cdot \ln(2.5) \cdot 2.5^{-3x^2} \cdot \frac{d}{dx}[-3x^2] \\ &=10 \cdot (0.9162) \cdot 2.5^{-3x^2} \cdot [-6x] \\ &=-55x \cdot 2.5^{-3x^2} \end{align*}$$
  ::dydx=ddx[10=2.5-3x2]=10ddx[2.5-33x2]=10n(2.5)=2.5-3x2]=10n(2.5)=2.5-33x2x2-dddx[-3x2]=10&(0.9162)=2.5-3x2][-3x2}[-6x]]\\\55x=2.5-3x2

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  Therefore, $\begin{align*}\frac{dy}{dx}=-55x \cdot 2.5^{-3x^2}\end{align*}$
  ::因此,Duddx55x2.5-3x2

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  Example 5
  ::例5

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  Given $\begin{align*}y=500 \cdot e^{-2x} \cdot \cos(5 \pi x)\end{align*}$ , what is $\begin{align*}\frac{dy}{dx}\end{align*}$ ?
  ::鉴于y=500e-2xcos(5xx),什么是?

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  $$\begin{align*}\frac{dy}{dx} &=\frac{d}{dx}[500 \cdot e^{-2x} \cdot \cos (5 \pi x)] \\ &=500 \cdot \left[\frac{d}{dx}(e^{-2x}) \cdot \cos (5 \pi x)+e^{-2x} \cdot \frac{d}{dx}(\cos (5 \pi x)) \right] && .... \text{Product Rule} \\ &=500 \cdot [\ln (e) \cdot e^{-2x} \frac{d}{dx}(-2x) \cdot \cos (5 \pi x)+e^{-2x} \cdot (-\sin(5 \pi x) \cdot \frac{d}{dx}(5 \pi x))] && .... \text{Use Chain Rule} \\ &=500 \cdot [(1) \cdot e^{-2x} \cdot (-2) \cdot \cos(5 \pi x)+e^{-2x} \cdot(-\sin(5 \pi x) \cdot 5 \pi)] && .... \text{Simplify} \\ &=-500 \cdot e^{-2x}[2 \cdot \cos(5 \pi x)+5 \pi \sin(5 \pi x)] && .... \text{Simplify} \end{align*}$$
  ::-=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

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  Therefore, $\begin{align*}\frac{dy}{dx}=-500 \cdot e^{-2x}[2 \cdot \cos(5 \pi x)+5 \pi \sin(5 \pi x)]\end{align*}$ .
  ::因此,Dudx500e-2x[2cos(5xx)+5sin(5xx)]。

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  Review
  ::回顾

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  For #1-14, find the derivative.
  ::为1 -14,找到衍生物。

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  1. $\begin{align*}y=7^x\end{align*}$
     ::y=7x y=7x
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  2. $\begin{align*}y=3^{2x}\end{align*}$
     ::y=32x y=32x
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  3. $\begin{align*}y=5^x-3x^2\end{align*}$
     ::y= 5x- 3x2 y= 5x- 3x2
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  4. $\begin{align*}y=2^{x^2}\end{align*}$
     ::y=2x2 y=2x2
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  5. $\begin{align*}y=e^{x^2}\end{align*}$
     ::y=ex2 y=ex2
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  6. $\begin{align*}f(x)=\frac{1}{\sqrt{\pi \sigma}}e^{-\alpha k(x-x_0)^2}\end{align*}$ where $\begin{align*}\sigma\end{align*}$ , $\begin{align*}\alpha\end{align*}$ , $\begin{align*}x_0\end{align*}$ , and $\begin{align*}k\end{align*}$ are constants and $\begin{align*}\sigma \ne 0\end{align*}$ .
     ::f(x) = 1 ek(x-x0) 2, 其中 、 α、 x0 和 k 是常数 和 0 。
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  7. $\begin{align*}y=e^{6x}\end{align*}$
     ::y=e6x y=e6x
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  8. $\begin{align*}y=e^{3x^3-2x^2+6}\end{align*}$
     ::y= e3x3 - 2x2+6 y= e3x3 - 2x2+6
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  9. $\begin{align*}y=\frac{e^x-e^{-x}}{e^x+e^{-x}}\end{align*}$
     ::y=ex- ex- ex- xex+e- x
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  10. $\begin{align*}y=\cos(e^x)\end{align*}$
      ::y=cos( ex) y=cos( ex)
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  11. $\begin{align*}y=e^{-x}{3^x}\end{align*}$
      ::y=e-x3x y=e-x3x
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  12. $\begin{align*}y=3^{{-x^2}+2x+1}\end{align*}$
      ::y=3 - x2+2x+1
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  13. $\begin{align*}y=2^x 3^x\end{align*}$
      ::y= 2x3xx
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  14. $\begin{align*}y=e^{-x} \sin x\end{align*}$
      ::y=e-xsinx y=e-xsinx
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  15. Find an equation of the tangent line to $\begin{align*}f(x)=x^3+2e^x\end{align*}$ at the point (0, 2).
      ::在点( 0, 2) 找到正切线到 f( x) =x3+2ex 的方程式 。

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  Review (Answers)
  ::回顾(答复)

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  ::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 。