3 种获取衍生品的方法

导数是一个算子，它计算一个量的瞬时变化率，通常是一个斜率。导数可用于获取有关函数的有用特征，例如其极值和根。从定义中找到导数可能很乏味，但有很多技术可以绕过它并更容易地找到导数。

脚步

方法 1 of 3：预赛

步骤 1. 理解导数的定义。

虽然这几乎永远不会用于实际获取衍生品，但对这个概念的理解仍然至关重要。

回想一下线性函数的形式是 y=mx+b.{displaystyle y=mx+b.}

To find the slope m{displaystyle m}

of this function, two points on the line are taken, and their coordinates are plugged into the relation m=y2−y1x2−x1.{displaystyle m={frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}

Of course, this can only be used with linear graphs.
For nonlinear functions, the line will be curved, so taking the difference of two points can only give the average rate of change between them. The line that intersects these two points is called the secant line, with a slope m=f(x+Δx)−f(x)Δx, {displaystyle m={frac {f(x+\Delta x)-f(x)}{Delta x}}, }

where Δx=x2−x1{displaystyle \Delta x=x_{2}-x_{1}}

is the change in x, {displaystyle x, }

and we have replaced y{displaystyle y}

with f(x).{displaystyle f(x).}

This is the same equation as the one before.

The concept of the derivatives comes in when we take the limit Δx→0.{displaystyle \Delta x\to 0.}

When this happens, the distance between the two points shrinks, and the secant line better approximates the rate of change of the function. When we do send the limit to 0, we end up with the instantaneous rate of change and obtain the slope of the tangent line to the curve (see animation above). Then, we end up with the definition of the derivative, where the prime symbol denotes the derivative of the function f.{displaystyle f.}
- f′(x)=limΔx→0f(x+Δx)−f(x)Δx{displaystyle f^{prime }(x)=\lim _{Delta x\to 0}{frac {f(x+\Delta x)-f(x)}{Delta x}}}
Finding the derivative from this definition stems from expanding the numerator, canceling, and then evaluating the limit, since immediately evaluating the limit will give a 0 in the denominator.

步骤 2. 理解导数符号。

导数有两种常见的表示法，但还有其他的表示法。

拉格朗日符号。

在上一步中，我们使用这个符号来表示函数 f(x){displaystyle f(x)} 的导数

by adding a prime symbol.
- f′(x){displaystyle f^{prime }(x)}
  
  $f^{{prime }}(x) /></p> </li></p> <li>这个符号是发音的$

${frac {{mathrm {d}}f}{{mathrm {d}}x}} /></p> </li></p> <li>（对于较短的表达式，函数可以放在分子中。）这个符号的字面意思是$

${frac {{mathrm {d}}^{{2}}f}{{mathrm {d}}x^{{2}}}}, /></p> <p> 其中 this 代表二阶导数。</li></p> <li>（注意有$

取导数第 3 步

步骤 1. 代入 (x+Δx){displaystyle (x+\Delta x)}

into the function.

For this example, we will define f(x)=2x2+6x.{displaystyle f(x)=2x^{2}+6x.}

f(x+Δx)=2(x+Δx)2+6(x+Δx)=2(x2+2xΔx+(Δx)2)+6x+6Δx=2x2+4xΔx+2(Δx)2+6x+6Δx.{displaystyle {begin{aligned}f(x+\Delta x)&=2(x+\Delta x)^{2}+6(x+\Delta x)\\&=2(x^{2}+2x\Delta x+(Delta x)^{2})+6x+6\Delta x\\&=2x^{2}+4x\Delta x+2(Delta x)^{2}+6x+6\Delta x.\end{aligned}}}

步骤 2. 将函数代入极限。

然后评估极限。

ddxf(x)=limΔx→0(2x2+4xΔx+2(Δx)2+6x+6Δx)−(2x2+6x)Δx=limΔx→04xΔx+2(Δx)2+6ΔxΔx=limΔx→0Δx(4x+2Δx) +6)Δx=limΔx→04x+2Δx+6=4x+6.{displaystyle {begin{aligned}{frac {mathrm {d} }{mathrm {d} x}}f(x)& =\lim _{Delta x\to 0}{frac {(2x^{2}+4x\Delta x+2(Delta x)^{2}+6x+6\Delta x)-(2x^ {2}+6x)}{Delta x}}\&=\lim _{Delta x\to 0}{frac {4x\Delta x+2(Delta x)^{2}+6\ Delta x}{Delta x}}\&=\lim _{Delta x\to 0}{frac {Delta x(4x+2\Delta x+6)}{Delta x}}\ &=\lim _{Delta x\to 0}4x+2\Delta x+6\&=4x+6.\end{aligned}}}
This is a lot of work for such a simple function. We will see that there are plenty of derivative rules to skirt past this type of evaluation.
You can find the slope anywhere on the function f(x)=2x2+6x.{displaystyle f(x)=2x^{2}+6x.}

Simply plug in any x value into the derivative df(x)dx=4x+6.{displaystyle {frac {mathrm {d} f(x)}{mathrm {d} x}}=4x+6.}

The Power Rule

步骤 1. 当 f(x){displaystyle f(x)}

is a polynomial function of degree n.

Multiply the exponent with the coefficient and bring down the power by one.

The formula is ddx(xn)=nxn−1.{displaystyle {frac {mathrm {d} }{mathrm {d} x}}(x^{n})=nx^{n-1}.}
Although the intuitive method seems to only apply to natural number exponents, it can be generalized to all real numbers; that is, n∈R.{displaystyle n\in \mathbb {R}.}

步骤 2. 使用前面的示例。

f(x)=2x2+6x.{displaystyle f(x)=2x^{2}+6x.}

Remember that x=x1.{displaystyle x=x^{1}.}

f(x)=2x2+6xddxf(x)=(2)2x2−1+(1)6x1−1=4x+6.{displaystyle {begin{aligned}f(x)&=2x^{2}+6x\\{frac {mathrm {d} }{mathrm {d} x}}f(x)&=(2)2x^{2-1}+(1)6x^{1-1}\\&=4x+6.\end{aligned}}}
We have used the property that the derivative of a sum is the sum of the derivatives (technically, the reason why we can do this is because the derivative is a linear operator). Obviously, the power rule makes finding derivatives of polynomials much easier.
Before going on, it is important to note that the derivative of a constant is 0, because the derivative measures the rate of change, and no such change exists with a constant.

Higher Order Derivatives

步骤 1. 再次区分。

取函数的高阶导数意味着您取导数的导数（2 阶）。例如，如果它要求您取三阶导数，只需将函数微分三次即可。对于 n 次多项式函数， {displaystyle n, }

the n+1{displaystyle n+1}

order derivative will be 0.

步骤 2. 取前面例子的三阶导数 f(x)=2x2+6x{displaystyle f(x)=2x^{2}+6x}

ddxf(x)=4x+6d2dx2f(x)=4d3dx3f(x)=0{displaystyle {begin{aligned}{frac {mathrm {d} }{mathrm {d} x}}f(x)&=4x+6\\{frac {mathrm {d} ^{2}}{mathrm {d} x^{2}}}f(x)&=4\\{frac {mathrm {d} ^{3}}{mathrm {d} x^{3}}}f(x)&=0\end{aligned}}}
In most applications of derivatives, especially in physics and engineering, you will at most differentiate twice, or perhaps three times.

The Product and Quotient Rules

步骤 1. 有关乘积规则的完整处理，请参阅本文。

一般来说，产品的导数不是等于衍生品的乘积。相反，每个功能“轮到”进行区分。

ddx(fg)=dfdxg+fdgdx{displaystyle {frac {mathrm {d} }{mathrm {d} x}}(fg)={frac {mathrm {d} f}{mathrm {d } x}}g+f{frac {mathrm {d} g}{mathrm {d} x}}}

步骤 2. 使用商法则求有理函数的导数。

与一般的乘积一样，商的导数确实不是等于导数的商。

ddx(fg)=gdfdx−fdgdxg2{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\left({frac {f}{g}}\right)={frac {g{frac {mathrm {d} f}{mathrm {d} x}}-f{frac {mathrm {d} g}{mathrm {d} x}}}{g^{2 }}}}

取导数第 11 步

步骤 1. 对嵌套函数使用链式法则。

例如，考虑 z(y){displaystyle z(y)}

is a differentiable function of y{displaystyle y}

and y(x){displaystyle y(x)}

is a differentiable function of x.{displaystyle x.}

Then there is a composite function z(y(x)), {displaystyle z(y(x)), }

or z{displaystyle z}

as a function of x, {displaystyle x, }

that we can take the derivative of.
- ddxz(y(x))=dzdydydx{displaystyle {frac {mathrm {d} }{mathrm {d} x}}z(y(x))={frac {mathrm {d} z}{mathrm {d} y}}{frac {mathrm {d} y}{mathrm {d} x}}}
  
  取导数第 12 步
  
  步骤 2. 考虑函数 f(x)=(2x4−x)3{displaystyle f(x)=(2x^{4}-x)^{3}}
  
  Notice that this function can be decomposed into two elementary functions, g(x)=2x4−x{displaystyle g(x)=2x^{4}-x}
  
  and h(g)=g3.{displaystyle h(g)=g^{3}.}
  
  Then, we want to find the derivative of the composition f(x)=h(g(x)).{displaystyle f(x)=h(g(x)).}
  - Use the chain rule ddxh(g(x))=dhdgdgdx.{displaystyle {frac {mathrm {d} }{mathrm {d} x}}h(g(x))={frac {mathrm {d} h}{mathrm {d} g}}{frac {mathrm {d} g}{mathrm {d} x}}.}
    
    We have now written the derivative in terms of derivatives that are easier to take. Then,
  - ddxh(g(x))=3(2x4−x)2(8x3−1).{displaystyle {frac {mathrm {d} }{mathrm {d} x}}h(g(x))=3(2x^{4}-x)^{2}(8x^{3}-1).}
    
    取导数第 13 步
    
    步骤 1. 有关隐微分的完整处理，请参阅本文。
    
    为了隐式区分，必须理解链式法则。
    
    取导数第 14 步
    
    步骤 2. 有关微分指数函数的完整处理，请参阅本文。
    - ddxex=ex{displaystyle {frac {mathrm {d} }{mathrm {d} x}}e^{x}=e^{x}}
    - ddxax=axln⁡a{displaystyle {frac {mathrm {d} }{mathrm {d} x}}a^{x}=a^{x}\ln a}
    - ddxln⁡x=1x{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\ln x={frac {1}{x}}}
    - ddxloga⁡x=1xln⁡a{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\log _{a}x={frac {1}{x\ln a}}}
    取导数第 15 步
    
    步骤 3. 记住基本的三角函数导数以及如何推导它们。
    - ddxsin⁡x=cos⁡x{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\sin x=\cos x}
    - ddxcos⁡x=−sin⁡x{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\cos x=-\sin x}
    - ddxtan⁡x=sec2⁡x{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\tan x=\sec ^{2}x}
    - ddxcot⁡x=−csc2⁡x{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\cot x=-\csc ^{2}x}
    - ddxsec⁡x=sec⁡xtan⁡x{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\sec x=\sec x\tan x}
    - ddxcsc⁡x=−csc⁡xcot⁡x{displaystyle {frac {mathrm {d} }{mathrm {d} x}}\csc x=-\csc x\cot x}
    Method 3 of 3: Using a Calculator
    
    Step 1. Press Alpha F2
    
    This will open the “Window” key, where you’ll see lots of options. Scroll over to the FUNC tab if you aren’t there already.
    
    These instructions are for new models of the TI-84 and the TI-84 Plus. Older models may be slightly different
    
    Step 2. Select nDeriv
    
    It’s the third option on the list. When you get to it, you can press “enter” to select it.
    
    Step 3. Enter your formula into the equation
    
    When you hit the derivative option, your calculator will give you a blank equation that looks like this: (d/d[])([])|x=[]{displaystyle (d/d[])([])|x=[]}
    
    . Go ahead and enter your specific numbers into the equation.
    - For example, if you were finding the derivative of the function x2{displaystyle x^{2}}
      
      where x=2{displaystyle x=2}
      
      , you’d enter (d/dx)(x2)|x=2{displaystyle (d/dx)(x^{2})|x=2}
      ${displaystyle (d/dx)(x^{2})|x=2} /></p> <p>.</li></p> <li>如果您在计算器的 Y 图中绘制了一个方程，则可以通过按 vars > Y-VARS > Function 将其输入到空白字段中。</li></p> </ul></p> <h4>第 4 步。点击“输入”以查找导数。</h4></p> <p>输入所有数字后，您可以在计算器上选择“输入”以获得答案。它将（希望）以易于理解的整数形式为您提供答案。</p></p> <h3>例如，在上面的等式中，导数为 4。</h3></p> <h2>视频 - 通过使用此服务，某些信息可能会与 YouTube 共享。</h2></p> <iframe 源代码=$

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