Differentiation of Exponential and Logarithmic Functions | 指数函数与对数函数的微分方法

📚 Differentiation of Exponential and Logarithmic Functions | 指数函数与对数函数的微分方法

In A-Level Mathematics, mastering the differentiation of exponential and logarithmic functions is essential for solving a wide range of calculus problems. This topic builds on the chain rule, product rule, and quotient rule, and opens the door to modelling real-world growth and decay processes.

在 A-Level 数学中，掌握指数函数与对数函数的微分是解决各类微积分问题的基础。这一主题建立在链式法则、乘积法则和商法则之上，并为我们建立现实世界中增长与衰减过程的模型铺平了道路。

---

1. The Derivative of eˣ | eˣ 的导数

The exponential function eˣ is unique in calculus because its derivative is itself. This property makes it the most important exponential function in mathematics.

指数函数 eˣ 在微积分中非常独特，因为它的导数就是它自身。这一性质使它成为数学中最重要的指数函数。

d/dx (eˣ) = eˣ

For example, if y = eˣ, then dy/dx = eˣ. This simple result is the foundation for differentiating any exponential expression.

例如，若 y = eˣ，则 dy/dx = eˣ。这个简单的结果是整个指数函数微分的基础。

---

2. The Derivative of aˣ | aˣ 的导数

For a general exponential function aˣ, where a > 0 and a ≠ 1, the derivative includes an extra factor of ln a.

对于一般的指数函数 aˣ（其中 a > 0 且 a ≠ 1），其导数包含一个额外的因子 ln a。

d/dx (aˣ) = aˣ ln a

This formula can be derived by writing aˣ as e^{x ln a}, then applying the chain rule. For example, d/dx (2ˣ) = 2ˣ ln 2.

该公式可以通过将 aˣ 写成 e^{x ln a}，然后应用链式法则来推导。例如，d/dx (2ˣ) = 2ˣ ln 2。

---

3. The Derivative of ln x | ln x 的导数

The natural logarithm function ln x is the inverse of eˣ. Its derivative is a simple reciprocal function.

自然对数函数 ln x 是 eˣ 的反函数。它的导数是一个简单的倒数函数。

d/dx (ln x) = 1/x, for x > 0

This result is very useful when dealing with expressions that are easier to differentiate after taking logarithms.

这一结果在遇到“先取对数再求导会更简单”的表达式时非常有用。

---

4. The Derivative of logₐx | logₐx 的导数

For a general logarithm with base a, the derivative is similar to ln x but includes a factor of 1/ln a.

对于底数为 a 的一般对数函数，其导数与 ln x 类似，但包含因子 1/ln a。

d/dx (logₐx) = 1/(x ln a), for x > 0, a > 0, a ≠ 1

logₐx = ln x / ln a, this derivative follows immediately from the constant multiple rule.

利用换底公式 logₐx = ln x / ln a，即可立刻由常数倍法则得到该导数。

---

5. The Chain Rule with Exponential Functions | 指数函数的链式法则

When the exponent is a function of x, we must use the chain rule. For y = e^u, where u = f(x), the derivative is:

当指数是 x 的函数时，我们必须使用链式法则。对于 y = e^u（其中 u = f(x)），其导数为：

d/dx (e^u) = e^u × du/dx

For example, d/dx (e^3x) = 3e^3x. More generally, d/dx (e^{sin x}) = cos x · e^{sin x}.

例如，d/dx (e^3x) = 3e^3x。更一般地，d/dx (e^{sin x}) = cos x · e^{sin x}。

---

6. The Chain Rule with Logarithmic Functions | 对数函数的链式法则

For y = ln u, where u = f(x), the chain rule gives:

对于 y = ln u（其中 u = f(x)），链式法则给出：

d/dx (ln u) = (1/u) × du/dx

Thus, d/dx (ln (2x + 1)) = 2/(2x + 1). A common alternative is d/dx (ln |x|) = 1/x, which holds for all x ≠ 0.

因此，d/dx (ln (2x + 1)) = 2/(2x + 1)。另一个常见形式是 d/dx (ln |x|) = 1/x，它对所有 x ≠ 0 都成立。

---

7. Product Rule with Exponentials and Logarithms | 乘积法则在指数和对数中的应用

of functions, the product rule states d/dx (uv) = u dv/dx + v du/dx. For example, y = x eˣ gives:

当一个表达式是函数的乘积时，乘积法则为 d/dx (uv) = u dv/dx + v du/dx。例如，y = x eˣ 有：

dy/dx = eˣ + x eˣ = eˣ (1 + x)

Similarly, for y = x² ln x, we have dy/dx = x + 2x ln x = x(1 + 2 ln x).

类似地，对于 y = x² ln x，我们有 dy/dx = x + 2x ln x = x(1 + 2 ln x)。

---

8. Quotient Rule with Exponentials and Logarithms | 商法则在指数和对数中的应用

(v du/dx - u dv/dx) / v². This is useful when a function is a ratio of two differentiable functions.

商法则为 d/dx (u/v) = (v du/dx - u dv/dx) / v²。当一个函数是两个可导函数之比时，该法则非常有用。

For example, differentiate y = eˣ / (x² + 1). Let u = eˣ and v = x² + 1. Then:

例如，求 y = eˣ / (x² + 1) 的导数。令 u = eˣ，v = x² + 1。则：

dy/dx = [(x² + 1)eˣ - eˣ(2x)] / (x² + 1)² = eˣ(x² - 2x + 1) / (x² + 1)²

---

广告 / Sponsored

9. Implicit Differentiation | 隐函数微分

Sometimes y is defined implicitly by an equation involving exponentials and logarithms. Differentiate both sides with respect to x, remembering to multiply by dy/dx when differentiating y.

有时 y 是由包含指数和对数的方程隐式定义的。对等式两边关于 x 求导时，记得在求含 y 的项时乘上 dy/dx。

y = 2x + 1, then differentiating gives (1/y) dy/dx = 2, so dy/dx = 2y. Substituting y = e^2x+1 yields dy/dx = 2e^2x+1.

例如，若 ln y = 2x + 1，两边求导得到 (1/y) dy/dx = 2，因此 dy/dx = 2y。代入 y = e^2x+1 得到 dy/dx = 2e^2x+1。

---

10. Logarithmic Differentiation | 对数微分法

tiation is a powerful technique used when differentiating functions of the form y = [f(x)]^g(x) or long products and quotients.

对数微分法是一种强有力的技巧，常用于形式为 y = [f(x)]^g(x) 的函数，或很长的乘积与商式。

Let y = xˣ. Take natural logarithms: ln y = x ln x. Differentiate implicitly:

设 y = xˣ。两边取自然对数：ln y = x ln x。然后隐式求导：

(1/y) dy/dx = ln x + 1 ⇨ dy/dx = xˣ (ln x + 1)

This method avoids using complicated rules and works well for many non-standard exponential forms.

这种方法避免使用复杂的法则，并且对许多非标准指数形式都很有效。

---

11. Common Exam Traps | 常见考试陷阱

• Forgetting the factor ln a when differentiating aˣ. The derivative is aˣ ln a, not just aˣ.

  对 aˣ 求导时忘记因子 ln a。导数应为 aˣ ln a，而不仅仅是 aˣ。

• Using the power rule for exponential functions. The power rule applies only to xⁿ, not to aˣ.

  对指数函数误用幂法则。幂法则只适用于 xⁿ，不适用于 aˣ。

• Omitting the domain condition x > 0 for ln x. The derivative 1/x is valid only where the original function is defined.

  忽略 ln x 的定义域条件 x > 0。导数 1/x 仅在原函数有定义的地方成立。

• Forgetting the chain rule factor when differentiating e^kx or ln (kx).

  对 e^kx 或 ln (kx) 求导时忘记链式法则中的因子。

• Misplacing signs in the quotient rule. Always write v du/dx - u dv/dx in the numerator.

  在商法则中弄错符号。分子中一定要写成 v du/dx - u dv/dx。

---

12. Summary and Revision Tips | 总结与复习建议

The key formulas to memorise are:

需要牢记的核心公式如下：

Function | 函数	Derivative | 导数
eˣ	eˣ
aˣ	aˣ ln a
ln x	1/x
logₐx	1/(x ln a)
eu(x)	eu(x) · u'(x)
ln (u(x))	u'(x)/u(x)

ln x, which is much easier to differentiate. Always check whether an expression can be simplified before applying a complicated rule.

练习在求导前利用对数运算法则重写表达式。例如，ln (x²) = 2 ln x，这样求导就简单得多。在应用复杂法则之前，务必检查表达式能否先化简。

Finally, remember that exponential and logarithmic functions are inverses of each other. Understanding this relationship helps you remember their derivative formulas and avoid confusion in exam questions.

最后，请记住指数函数和对数函数互为反函数。理解这一关系有助于记忆它们的导数公式，并在考试中避免混淆。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）