📚 10F Calculus applied to more functions | 10F 微积分应用于更多函数
In earlier calculus topics, we explored differentiation and integration of polynomial functions. However, the real power of calculus emerges when we extend these operations to a broader class of functions: exponentials, logarithms, and trigonometric functions. This article covers the essential rules for differentiating and integrating these functions, along with the application of chain, product, and quotient rules. Mastery of these concepts is crucial for success in IB Mathematics, as they form the basis for modelling growth, decay, oscillations, and many real-world phenomena.
在前面的微积分专题中，我们探讨了多项式函数的求导与积分。然而，微积分的真正威力在于当我们将这些运算扩展到更广泛的函数类别时：指数函数、对数函数和三角函数。本文涵盖了这些函数微分的核心法则以及链式法则、乘积法则和商法则的应用，同时介绍它们的积分。掌握这些内容对于 IB 数学取得好成绩至关重要，因为它们为建立增长、衰减、振荡等许多现实世界现象的模型奠定了基础。
1. Beyond Polynomials: A New Family of Functions | 超越多项式：一类新的函数
So far, the functions we differentiated were mainly powers of x. Now we introduce exponential functions like eˣ and 2ˣ, logarithmic functions like ln x, and circular functions like sin x and cos x. Each type has its own unique derivative behaviour, often involving the function itself or simple reciprocals. Understanding these derivatives allows us to tackle more complex problems in kinematics, population dynamics, and geometry.
到目前为止，我们求导的函数主要是 x 的幂函数。现在我们要引入诸如 eˣ 和 2ˣ 的指数函数、ln x 这样的对数函数以及 sin x 和 cos x 等圆函数。每种函数都有其独特的导数特性，往往涉及函数自身或简单的倒数。理解这些导数让我们能够解决运动学、种群动态和几何学中更复杂的问题。
Equally important are the integrals of these functions, which reverse the differentiation process and give us tools to compute areas and accumulated change. We will build up a comprehensive table of derivatives and antiderivatives, and then practise applying them in combination with the differentiation rules you already know.
同样重要的是这些函数的积分，它逆转了微分过程，为我们提供了计算面积和累积变化的工具。我们将构建一个全面的导数与反导数表，然后练习将它们与你已经掌握的微分法则相结合。
2. Derivative of the Exponential Function eˣ | 指数函数 eˣ 的导数
The natural exponential function eˣ is remarkable because its derivative is itself. This property makes it fundamental in calculus. The formal rule is:
自然指数函数 eˣ 非常特别，因为它的导数就是它本身。这一性质使其成为微积分的基础。正式的求导法则是：
d/dx (eˣ) = eˣ
When we apply the chain rule, the derivative of ekx becomes k ekx. For example, d/dx (e2x) = 2 e2x and d/dx (e-x) = -e-x ay. The constant e (approximately 2.718) is chosen precisely so that the gradient of y = eˣ at any point equals the function value.
当我们应用链式法则时，ekx 的导数为 k ekx。例如，d/dx (e2x) = 2 e2x，d/dx (e-x) = -e-x。这种规律对于求解微分方程以及分析指数增长或衰减至关重要。常数 e（约等于 2.718）正是为了使 y = eˣ 在任意点的斜率都等于函数值而选取的。
3. Derivative of aˣ and the Chain Rule with Exponentials | aˣ 的导数以及指数函数的链式法则
For any positive base a (a ≠ 1), the derivative of aˣ is aˣ ln a. This can be derived by rewriting aˣ as ex ln a. So, d/dx (aˣ) = aˣ ln a. When combined with the chain rule, d/dx (a^(u(x))) = a^(u(x)) ln a · u'(x). For instance, d/dx (23x) = 23x ln 2 · 3.
对于任意正数底 a (a ≠ 1)，aˣ 的导数为 aˣ ln a。这可以通过将 aˣ 重写为 ex ln a 推导得出。因此，d/dx (aˣ) = aˣ ln a。结合链式法则，d/dx (a^(u(x))) = a^(u(x)) ln a · u'(x)。例如，d/dx (23x) = 23x ln 2 · 3。
| f(x) | f'(x) |
| eˣ | eˣ |
| ekx | k ekx |
| aˣ | aˣ ln a |
| akx | k akx ln a |
Remember that the factor ln a is simply a constant, so it behaves exactly like any other coefficient when using the product or chain rules. Always check that the base a is positive and not equal to 1, as otherwise the function is constant or undefined.
记住因子 ln a 只是一个常数，因此在应用乘积法则或链式法则时，它的表现与其他系数
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图