微积分 | Derivative rules

注：本文为 “微积分 | Derivative rules” 相关。
英文引文，机翻未校。
如有内容异常，请看原文。

Derivative rules – Common Rules, Explanations, and Examples

导数规则——常见规则、解释及示例

Having a list of derivative rules you can always go back to will make your learning of differential calculus topics much easier. These derivative rules are the most fundamental rules you’ll encounter, and knowing how to apply them to differentiate different functions is crucial in calculus and its fields of applications.
拥有一份可以随时查阅的导数规则列表，将使你学习微分学相关主题变得更加轻松。这些导数规则是你将遇到的最基本的规则，掌握如何应用它们来对不同函数求导，在微积分及其应用领域至关重要。

Mastering the fundamental derivative rules will help you in differentiating complex functions and deriving more complex derivative rules.
掌握这些基本的导数规则，将帮助你对复杂函数求导，并推导出更复杂的导数规则。

This article will review all the fundamental derivative rules we’ve learned in the past and see how we can combine different rules to find the derivative of functions with multiple terms. This will also serve as a refresher and make sure you understand the basic derivative rules before learning more complex derivative rules.
本文将回顾我们过去学习的所有基本导数规则，并探讨如何结合不同规则来求解包含多个项的函数的导数。这也将作为一个复习，确保你在学习更复杂的导数规则之前，已经理解了这些基本的导数规则。

What are the common rules of derivatives?

常见导数规则有哪些？

In the past, we’ve derived the rules from the fundamental definition of derivatives, as shown below.
过去，我们从导数的基本定义出发，推导出了这些规则，如下所示。

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

As we have also observed, this process can get tedious, especially when you’re working with functions with multiple terms, complex forms, and composite functions. When dealing with these types of functions’ derivatives, it helps if we already know the common rules of derivatives by heart.
正如我们所观察到的，这一过程可能相当繁琐，尤其是当你处理包含多个项、复杂形式以及复合函数的函数时。在处理这些函数的导数时，如果已经牢记常见的导数规则，将会非常有帮助。

We’ll review each fundamental rule here and briefly discuss how we can apply the particular derivative rule.
我们将在这里回顾每一条基本规则，并简要讨论如何应用每一条特定的导数规则。

– Constant Rule: 常数规则

$\frac{d}{dx}c=0$

The constant rule states that the derivative of a constant is equal to 0. This means that when you’re given a polynomial function, the constants’ derivatives will be equal to 0 using this rule.
常数规则表明，常数的导数等于 0。这意味着在给定的多项式函数中，常数项的导数将等于 0。

• $\frac{d}{dx}100=0$
• $\frac{d}{dx}-1=0$
• $\frac{d}{dx}\pi =0$

– Constant Multiple Rule: 常数倍数规则

$\frac{d}{dx}[c\cdot f(x)]=c\cdot \frac{d}{dx}f(x)$

When we have a coefficient before an expression, we can simply factor out the coefficient and take the derivative of the remaining expression.
当一个表达式前面有一个系数时，我们可以直接提取出该系数，并对剩余的表达式求导。

• $\frac{d}{dx}3x=3\frac{d}{dx}x$
• $\frac{d}{dx}4x^2=4\frac{d}{dx}{x}^2$
• $\frac{d}{dx}-12e^x=-12\frac{d}{dx}{e}^x$

– Power Rule: 幂规则

$\frac{d}{dx}{x}^n=n{x}^{n-1}$

According to the power rule, differentiating a powered expression, $x^n$, we can simply use $n$ as the derivative’s coefficient and decrease the exponent of $x$ by 1.
根据幂规则，对幂表达式 $x^n$ 求导时，我们可以直接使用 $n$ 作为导数的系数，并将 $x$ 的指数减 1。

• $\frac{d}{dx}{x}^5=5x^4$
• $\frac{d}{dx}{x}^{14}=14x^{13}$
• $\frac{d}{dx}{x}^{-4}=-4x^{-5}$

– Sum and Difference Rules: 和差规则

$\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$

When a function is made up of two or more simpler functions, we can find its derivative by adding or subtracting the derivatives of the simpler functions.
当一个函数由两个或多个更简单的函数组成时，我们可以通过加或减这些简单函数的导数来求得其导数。

• $\frac{d}{dx}(x^5-2x)=\frac{d}{dx}{x}^5-\frac{d}{dx}2x$
• $\frac{d}{dx}(x^3+4x+6)=\frac{d}{dx}{x}^3+\frac{d}{dx}4x+\frac{d}{dx}6$
• $\frac{d}{dx}(x^2+x-\frac{1}{x})=\frac{d}{dx}{x}^2-\frac{d}{dx}x-\frac{d}{dx}\frac{1}{x}$

– Product Rule: 乘积规则

$\frac{d}{dx}[f(x)\cdot g(x)]=f^{\prime }(x)g(x)+g^{\prime }(x)f(x)$

The product rule states that when a function is a product of two functions, we can find the derivative of functions by pairing the derivative of the first function and the second function. Do the same for the second function’s derivative and first function.
乘积规则表明，当一个函数是两个函数的乘积时，我们可以通过将第一个函数的导数与第二个函数配对，以及将第二个函数的导数与第一个函数配对，来求得该函数的导数。

• $\frac{d}{dx}(4x)(e^x)=\frac{d}{dx}(4x)\cdot e^x-\frac{d}{dx}(e^x)\cdot 4x$
• $\frac{d}{dx}(-2x)(\sqrt{x})=\frac{d}{dx}(-2x)\cdot \sqrt{x}-\frac{d}{dx}\sqrt{x}\cdot (-2x)$
• $\frac{d}{dx}4x{e}^x=\frac{d}{dx}(4x)\cdot e^x-\frac{d}{dx}(e^x)\cdot 4x$

– Quotient Rule: 商规则

$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$

From the quotient rule, we can find the derivative of the two functions’ ratio by taking the difference between the product of the derivative of the numerator and the denominator minus the product of the derivative of the denominator and the numerator. Divide this result by the square of the denominator.
根据商规则，我们可以通过计算分子的导数与分母的乘积减去分母的导数与分子的乘积，再将结果除以分母的平方，来求得两个函数比值的导数。

• $\frac{d}{dx}\frac{3x^2}{\sin x}=(\sin x)\frac{d}{dx}(3x^2)-(3x^2)\frac{d}{dx}\sin x÷(\sin x{)}^2$
• $\frac{d}{dx}\frac{x^2}{x-1}=(2x-1)\frac{d}{dx}x-x\frac{d}{dx}(2x-1)÷(2x-1{)}^2$
• $\frac{d}{dx}\frac{3x^2-1}{e^x}=e^x\frac{d}{dx}(3x^2-1)-(3x^2-1)\frac{d}{dx}{e}^x÷(e^x{)}^2$

These are the fundamental derivative rules that we should learn by heart if we want to master differentiating the common functions we’ve been dealing with in the past.
这些是我们应该牢记的基本导数规则，以便掌握过去我们处理过的常见函数的求导方法。

How to differentiate functions using the list of derivative rules?

如何使用导数规则列表对函数求导？

Now that we’ve reviewed the fundamental derivative rules we need, the next important thing for us to learn is knowing how to use a combination of these rules to differentiate polynomial, rational, radical, and other fundamental functions.
现在我们已经复习了所需的这些基本导数规则，接下来需要学习的是如何结合使用这些规则来对多项式、有理函数、根式函数以及其他基本函数求导。

• See if the sum or difference rule applies to the function and apply them first.
  检查和差规则是否适用于该函数，并优先应用。

• When you find constants in the expressions, expect it’s the constants’ derivatives to become zero.
  当表达式中出现常数时，预期其导数将变为零。

• If the expressions have coefficients before the terms, make sure to apply the constant multiple rule.
  如果表达式中各项前面有系数，确保应用常数倍数规则。

• Check if the function is in factored form or a rational expression and apply the product or quotient rule, respectively.
  检查该函数是否为因式分解形式或有理表达式，并分别应用乘积规则或商规则。

There are instances when we might need to apply more complex derivatives, such as the formula for differentiating composite functions (also known as the chain rule) and trigonometric expressions, as shown below.
有时我们可能需要应用更复杂的导数，例如复合函数的求导公式（也称为链式法则）和三角表达式，如下所示。

Now that we’ve learned the important derivative rules and the process of differentiating complex expressions, it’s time for us to apply what we’ve learned by working on the functions shown below.
现在我们已经学习了重要的导数规则以及对复杂表达式求导的过程，是时候通过解决以下函数来应用我们所学到的知识了。

Example 1

Find the derivative of the polynomial function, $f(x)=4x^3-5x^2+6x-1$.
求多项式函数 $f(x)=4x^3-5x^2+6x-1$ 的导数。

Solution
解

Since we can see a polynomial function with terms being added and subtracted to each other, we can apply the sum and difference rules to find the derivative of $f(x)$. This means that we take the derivative of each of the terms to find $f^{\prime }(x)$.
由于我们看到的是一个包含相加和相减项的多项式函数，因此可以应用和差规则来求 $f(x)$ 的导数。这意味着我们需要分别对每一项求导，以得到 $f^{\prime }(x)$。

$$\begin{array}{rl}{f}^{\prime }(x)& =\frac{d}{dx}[4x^3-5x^2+6x-1]\\ & =\frac{d}{dx}4x^3-\frac{d}{dx}5x^2+\frac{d}{dx}6x-\frac{d}{dx}1\end{array}$$

We can then replace $\frac{d}{dx}1$ with 0 and apply the given constant multiple rule to take out the coefficients from each term.
然后，我们将 $\frac{d}{dx}1$ 替换为 0，并应用常数倍数规则，将每一项的系数提取出来。

$$\begin{array}{rl}{f}^{\prime }(x)& =\frac{d}{dx}4x^3-\frac{d}{dx}5x^2+\frac{d}{dx}6x-0\\ & =4\frac{d}{dx}{x}^3-5\frac{d}{dx}{x}^2+6\frac{d}{dx}x\end{array}$$

We can use the power rule for each term to differentiate powers $x-x^3$, $x^2$, and $x=x^1$.
我们可以对每一项使用幂规则来对 $x-x^3$、$x^2$ 和 $x=x^1$ 求导。

$$\begin{array}{rl}{f}^{\prime }(x)& =4(3x^{3-1})-5(2x^{2-1})+6(1x^{1-1})\\ & =12x^2-10x^1+6x^0=12x^2-10x+6\end{array}$$

This shows that through different derivative rules, we can easily find the derivative of polynomials such as $f(x)=4x^3-5x^2+6x-1$. In fact, we have $f^{\prime }(x)=12x^2-10x+6$.
这表明，通过使用不同的导数规则，我们可以轻松地求出像 $f(x)=4x^3-5x^2+6x-1$ 这样的多项式的导数。实际上，我们得到 $f^{\prime }(x)=12x^2-10x+6$。

Example 2

Find the derivative of the function, $g(x)=\sin x^4-5x^5+x$.
求函数 $g(x)=\sin x^4-5x^5+x$ 的导数。

Solution
解

We’ll now work with a more complex function, that consists of three terms: $\sin x^4$, $5x^4$, and $x$. Through sum and difference rules, we’ll be able to find the expression for $g^{\prime }(x)$ by finding the corresponding derivatives of the three terms.
现在我们将处理一个更复杂的函数，它包含三个项：$\sin x^4$、$5x^4$ 和 $x$。通过应用和差规则，我们可以通过求这三个项的导数来得到 $g^{\prime }(x)$ 的表达式。

$$\begin{array}{rl}{g}^{\prime }(x)& =\frac{d}{dx}[\sin x^4-5x^5+x]\\ & =\frac{d}{dx}\sin x^4-\frac{d}{dx}5x^5+\frac{d}{dx}x\end{array}$$

Each term will require different sets of derivative rules for us to find their derivatives, so here’s a table summarizing how we can differentiate each term.
每一项都需要不同的导数规则来求导，以下是一个总结如何对每一项求导的表格。

Let’s substitute these expressions back into $g^{\prime }(x)$ to find the derivative of $g(x)$.
现在，我们将这些表达式代入 $g^{\prime }(x)$，以求得 $g(x)$ 的导数。

$$\begin{array}{rl}{g}^{\prime }(x)& =\frac{d}{dx}\sin x^4-\frac{d}{dx}5x^5+\frac{d}{dx}x\\ & =\frac{1}{4}\cos x-25x^4+\frac{1}{2x}\end{array}$$

This example shows how we can combine the different derivative rules to find the derivative of a more complex function such as $g(x)$. Hence, we have $g^{\prime }(x)=\frac{1}{4}\cos x-25x^4+\frac{1}{2x}$.
这个例子展示了我们如何结合不同的导数规则来求得像 $g(x)$ 这样更复杂函数的导数。因此，我们得到 $g^{\prime }(x)=\frac{1}{4}\cos x-25x^4+\frac{1}{2x}$。

Practice Questions

练习题

1. Find the derivative of the following polynomial functions:

求以下多项式函数的导数：

a. $f(x)=12x^5-4x^6+2x^4-6$

b. $g(x)=-6x^5-12x^3-8$

c. $h(x)=-5x^8+6x^6-12x^2+9x$

2. Find the derivative of the following functions:

求以下函数的导数：

a. $f(x)=25x-6x^2$

b. $g(x)=4x^2-4x+\frac{1}{x}$

c. $h(x)=2x-5x^3+\frac{1}{x}$

3. Find the derivative of the following functions:

求以下函数的导数：

a. $f(x)=(x-1)(2x+3{)}^2$

b. $g(x)=\frac{x^2-4x+4}{3x}$

c. $h(x)=\frac{6x^2}{4}-3x$

Answer Key

答案

1.

a. $f^{\prime }(x)=60x^4-24x^5+8x^3$

b. $g^{\prime }(x)=-30x^4-36x^2$

c. $h^{\prime }(x)=-40x^7+36x^5-24x+9$

2.

a. $f^{\prime }(x)=-25x^2-12x$

b. $g^{\prime }(x)=8x-2x^{-1}{x}^2$

c. $h^{\prime }(x)=\frac{1}{x}-5\cdot 3x^{2/3}-\frac{1}{x^2}$

3.

a. $f^{\prime }(x)=(6x-1)(2x+3)$ or $f^{\prime }(x)=12x^2+6x-3$

b. $g^{\prime }(x)=\frac{1}{3}-\frac{4}{3x^2}$ or $g^{\prime }(x)=-\frac{4-x^2}{3x^2}$

c. $h^{\prime }(x)=\frac{48x-27x^2}{(4-3x{)}^{3/2}}$

via：

• Derivative rules - Common Rules, Explanations, and Examples
  https://www.storyofmathematics.com/derivative-rules/

  • Sum Rule - Everything You Need To Know!
    https://www.storyofmathematics.com/sum-rule/

  • Difference rule - Derivation, Explanation, and Example
    https://www.storyofmathematics.com/difference-rule/

• Integral Properties - Definition, Process, and Proof
  https://www.storyofmathematics.com/integral-properties/