Simple Rules of Differentiation | 简单微分法则

📚 Simple Rules of Differentiation | 简单微分法则

Differentiation is a cornerstone of calculus, providing a way to measure how a function changes as its input changes. In IB Mathematics, mastering the simple rules of differentiation allows you to quickly find derivatives without relying on the limit definition every time. These rules cover constants, power functions, basic exponential and logarithmic functions, and trigonometric functions. By learning them thoroughly, you will be able to differentiate a wide range of functions efficiently and apply them to problems such as finding tangents, velocities, and optimisation.

微分是微积分的基石，它提供了一种衡量函数随输入变化而变化的方式。在 IB 数学中，掌握简单的微分法则可以让你快速求导，而无需每次都依赖极限定义。这些法则涵盖了常数、幂函数、基本指数函数与对数函数以及三角函数。通过全面学习这些法则，你将能够高效地对各种函数求导，并将其应用于求切线、速率和优化等实际问题。

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1. Introduction to Differentiation and Notation | 微分简介与记号

Differentiation is the process of finding the derivative of a function. The derivative f' = f(x). There are two common notations: Leibniz notation (dy/dx) and Lagrange notation (f'(x)). For example, if f(x) = x², its derivative is f'(x) = 2x, or dy/dx = 2x.

微分是求函数导数的过程。导数 f'(x) 或 dy/dx 表示 y 随 x 变化的瞬时变化率。从几何上看，它给出了曲线 y = f(x) 上任意一点切线的斜率。常见的记号有两种：莱布尼茨记号（dy/dx）和拉格朗日记号（f'(x)）。例如，若 f(x) = x²，其导数为 f'(x) = 2x，或写作 dy/dx = 2x。

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2. Derivative of a Constant | 常数的导数

sense because a horizontal line has zero slope everywhere. In symbols, if f(x) = c, then f'(x) = 0 or d/dx (c) = 0. For instance, d/dx (5) = 0 and d/dx (-√2) = 0.

任何常数函数 c（c 为实数）的导数为零。这一规律很直观，因为水平线在任何位置的斜率都是零。用符号表示：若 f(x) = c，则 f'(x) = 0，或 d/dx (c) = 0。例如，d/dx (5) = 0，d/dx (-√2) = 0。

d/dx (c) = 0

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3. The Power Rule | 幂法则

The power rule is one of the most useful differentiation rules. It states that if f(x) = xⁿ, where n is any real number, then f'(x) = n x^n-1. Using Leibniz notation: d/dx (xⁿ) = n x^n-1. This rule works for positive integers, negative integers, and fractions. For example, d/dx (x³) = 3x²; d/dx (x^-2) = -2x^-3; and d/dx (√x) = d/dx (x^1/2) = (1/2)x^-1/2. The power rule is derived from the limit definition and greatly streamlines the differentiation of polynomial-like terms.

幂法则是最有用的微分法则之一。该法则指出：若 f(x) = xⁿ，其中 n 为任意实数，则 f'(x) = n x^n-1。使用莱布尼茨记号：d/dx (xⁿ) = n x^n-1。该法则适用于正整数、负整数和分数指数。例如，d/dx (x³) = 3x²；d/dx (x^-2) = -2x^-3；d/dx (√x) = d/dx (x^1/2) = (1/2)x^-1/2。幂法则由极限定义推导而来，大大简化了多项式类项的求导。

d/dx (xⁿ) = n x^n-1

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4. Constant Multiple Rule | 常数倍法则

constant. That is, if f(x) is differentiable and c is a constant, then d/dx [c f(x)] = c f'(x). This rule lets you 'factor out' constants when differentiating. For example, d/dx (7x⁴) = 7 * d/dx (x⁴) = 7 * 4x³ = 28x³. Similarly, d/dx (-3/x) = -3 * d/dx (x^-1) = -3 * (-x^-2) = 3/x².

若将函数乘以一个常数，导数也会乘以同一个常数。即，如果 f(x) 可导且 c 为常数，则 d/dx [c f(x)] = c f'(x)。该法则允许在求导时将常数“提出来”。例如，d/dx (7x⁴) = 7 * d/dx (x⁴) = 7 * 4x³ = 28x³。类似地，d/dx (-3/x) = -3 * d/dx (x^-1) = -3 * (-x^-2) = 3/x²。

d/dx [c f(x)] = c f'(x)

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5. Sum and Difference Rule | 和差法则

ly the sum or difference of their individual derivatives. Formally, d/dx [f(x) ± g(x)] = f'(x) ± g' h the previous ones, makes differentiating polynomials straightforward. Consider h(x) = 3x⁴ - 5x² + 2x - 7. Applying the sum/difference rule: h'(x) = d/dx(3x⁴) - d/dx(5x²) + d/dx(2x) - d/dx(7). Then using constant multiple and power rules: h'(x) = 12x³ - 10x + 2 - 0 = 12x³ - 10x + 2.

函数和或差的导数就是各函数导数的和或差。形式上，d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。这一法则结合前面的法则，能让多项式的求导变得非常简单。考虑 h(x) = 3x⁴ - 5x² + 2x - 7。应用和差法则：h'(x) = d/dx(3x⁴) - d/dx(5x²) + d/dx(2x) - d/dx(7)。然后利用常数倍法则和幂法则：h'(x) = 12x³ - 10x + 2 - 0 = 12x³ - 10x + 2。

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

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6. Derivative of the Exponential Function e^x | 指数函数 e^x 的导数

The natural exponential function e^x has the remarkable property that it is its own derivative. In other words, d/dx (e^x) = e^x e exponential function at x = 0 is exactly 1. No matter how many times you differentiate e^x, the result remains e^x. For example, if f(x) = 4e^x, then f'(x) = 4e^x by the constant multiple rule.

自然指数函数 e^x 具有一个显著特性，即其导数就是它自身。换句话说，d/dx (e^x) = e^x。这可以利用极限定义证明，并且自然底数 e ≈ 2.718 的定义恰好使得指数函数在 x = 0 处的斜率为 1。无论你对 e^x 求导多少次，结果始终是 e^x。例如，若 f(x) = 4e^x，则由常数倍法则得 f'(x) = 4e^x。

d/dx (e^x) = e^x

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7. Derivative of the Natural Logarithm ln x | 自然对数 ln x 的导数

s given by d/dx (ln x) = 1/x. This formula follows from the inverse relationship between e^x r instance, the slope of the curve y = ln x at x = 2 is 1/2, and at x = 1 it is exactly 1.

自然对数函数 f(x) = ln x（定义域为 x > 0）的导数为 d/dx (ln x) = 1/x。该公式源于 e^x 与 ln x 之间的反函数关系。这是一个关键结论，与指数函数的导数相结合，使我们能够处理许多超越函数。例如，曲线 y = ln x 在 x = 2 处的斜率为 1/2，而在 x = 1 处斜率恰好为 1。

d/dx (ln x) = 1/x, x > 0

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8. Derivatives of Trigonometric Functions | 三角函数的导数

sin x. Remember the negative sign for the cosine derivative. These simple rules, combined with the chain rule later, open the door to differentiating a vast array of trigonometric expressions.

两个基本的三角函数导数是正弦和余弦的导数。sin x 的导数是 cos x，而 cos x 的导数是 -sin x。这些公式可以利用极限和三角恒等式推导得出。使用记号：d/dx (sin x) = cos x，d/dx (cos x) = -sin x。请牢记余弦导数带有负号。这些简单法则与链式法则结合后，可以对大量的三角函数表达式进行求导。

d/dx (sin x) = cos x

d/dx (cos x) = -sin x

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9. Putting it Together: Differentiating Polynomials and More | 综合应用：多项式及其他函数求导

ants, powers, exponentials, logarithms, and sines/cosines. For example, consider f(x) = 2x⁵ - 3/x² + 4e^x - ln x + 7 sin x. Rewriting the rational term as -3x^-2, we apply the sum/difference rule: f'(x) = d/dx(2x⁵) - d/dx(3x^-2) + d/dx(4e^x) - d/dx(ln x) + d/dx(7 sin x). Then, using the appropriate rules:

掌握了这些简单法则，你就可以对包含常数、幂函数、指数函数、对数函数以及正弦余弦函数的组合表达式求导。例如，考虑 f(x) = 2x⁵ - 3/x² + 4e^x - ln x + 7 sin x。将分式项改写为 -3x^-2，应用和差法则：f'(x) = d/dx(2x⁵) - d/dx(3x^-2) + d/dx(4e^x) - d/dx(ln x) + d/dx(7 sin x)。然后利用相应法则：

• d/dx(2x⁵) = 10x⁴
• d/dx(3x^-2) = -6x^-3 (so subtracting gives +6x^-3)
• d/dx(4e^x) = 4e^x
• d/dx(ln x) = 1/x
• d/dx(7 sin x) = 7 cos x

Thus, the derivative is f'(x) = 10x⁴ + 6x^-3 + 4e^x - 1/x + 7 cos x. This illustrates how the simple rules work in harmony to differentiate complex-looking functions rapidly.

因此，导数为 f'(x) = 10x⁴ + 6x^-3 + 4e^x - 1/x + 7 cos x。这展示了简单法则如何协同工作，快速对看上去复杂的函数进行求导。

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10. Application: Equation of a Tangent Line | 应用：求切线方程

iven point. Given a function y = f(x) and a point x = a, the slope of the tangent is m = f'(a). The line passes through (a, f(a)), so using point-slope form: y - f(a) = f'(a)(x - a). For example, let f(x) = x³ - 2x + 1. Find the tangent at x = 1. First, f(1) = 1 - 2 + 1 = 0. The derivative is f'(x) = 3x² - 2. At x = 1, f' 3 - 2 = 1. Hence, the tangent line equation is y - 0 = 1(x - 1), simplified to y = x - 1.

微分的一个直接应用是求曲线在某一点处的切线方程。给定函数 y = f(x) 和点 x = a，切线的斜率为 m = f'(a)。该直线经过点 (a, f(a))，因此利用点斜式：y - f(a) = f'(a)(x - a)。例如，设 f(x) = x³ - 2x + 1，求 x = 1 处的切线。首先，f(1) = 1 - 2 + 1 = 0。导数为 f'(x) = 3x² - 2。在 x = 1 处，f'(1) = 3 - 2 = 1。因此，切线方程为 y - 0 = 1(x - 1)，化简得 y = x - 1。

These simple rules thus provide not only algebraic proficiency but also the foundation for solving geometric and physical problems involving rates of change.

因此，这些简单法则不仅提供了代数计算的能力，还为解决涉及变化率的几何与物理问题打下了基础。

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