AP Calculus AB & BC Core Formulas Summary | AP微积分AB与BC核心公式汇总

📚 AP Calculus AB & BC Core Formulas Summary | AP微积分AB与BC核心公式汇总

Welcome to the ultimate AP Calculus AB & BC formula sheet. This concise compilation covers all fundamental limits, derivatives, integrals, and series formulas required for the AP exams. Mastering these formulas is essential for success in both Calculus AB and BC, with BC-specific topics clearly indicated.

欢迎查阅AP微积分AB与BC终极公式表。这份简洁的汇编涵盖了AP考试所需的所有基本极限、导数、积分与级数公式。掌握这些公式对于AB和BC考生的成功至关重要，文中会清晰标明仅限BC的部分。

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1. Limits and Continuity | 极限与连续性

The limit of a function f(x) as x approaches c is L, written as lim (x→c) f(x) = L, provided the values of f(x) can be made arbitrarily close to L by taking x sufficiently close to c (but x ≠ c).

函数f(x)当x趋向c时的极限为L，记作lim (x→c) f(x) = L，如果当x充分接近c（但x≠c）时，f(x)的值可以任意接近L。

One-sided limits: lim (x→c⁻) f(x) and lim (x→c⁺) f(x) must both exist and be equal for the two-sided limit to exist.

单侧极限：lim (x→c⁻) f(x) 和 lim (x→c⁺) f(x) 必须都存在且相等，双侧极限才存在。

A function f is continuous at x = c if lim (x→c) f(x) = f(c). If this holds for all points in an interval, f is continuous on that interval. Polynomials, rational functions (where defined), exponential, logarithmic, and trigonometric functions are continuous on their domains.

如果lim (x→c) f(x) = f(c)，则函数f在x=c处连续。若在一个区间内每点都连续，则函数在该区间上连续。多项式、有理函数（在定义域内）、指数、对数、三角函数在其定义域上都是连续的。

Essential special limits:

必须掌握的特殊极限：

lim (x→0) sin x / x = 1

lim (x→0) (1 – cos x) / x = 0

lim (x→∞) (1 + 1/x)^x = e

Also remember limit laws: sum, product, quotient (provided denominator limit ≠ 0), and Squeeze Theorem.

同时牢记极限运算法则：和、积、商（分母极限不为0）以及夹逼定理。

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2. Basic Derivative Formulas | 基本导数公式

The derivative gives the instantaneous rate of change. The following table summarizes the most important elementary derivatives.

导数给出瞬时变化率。下表总结了最重要的初等函数导数。

Function	Derivative
c (constant)	0
xⁿ	n xⁿ⁻¹
aˣ	aˣ ln a
eˣ	eˣ
ln x	1/x, x > 0
logₐ x	1/(x ln a)

For absolute value, the derivative of |x| is piecewise: 1 for x > 0 and -1 for x < 0; it is not differentiable at x = 0.

对于绝对值函数，|x|的导数是分段函数：x > 0时为1，x < 0时为-1；在x=0处不可导。

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3. Differentiation Rules | 求导法则

Sum/Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

和差法则： d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Product Rule: d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)

积法则： d/dx [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)

Quotient Rule: d/dx [f(x) / g(x)] = [f'(x) g(x) – f(x) g'(x)] / [g(x)]², provided g(x) ≠ 0.

商法则： d/dx [f(x) / g(x)] = [f'(x) g(x) – f(x) g'(x)] / [g(x)]²，g(x) ≠ 0。

Chain Rule: If y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx), equivalently d/dx [f(g(x))] = f'(g(x)) · g'(x).

链式法则： 如果 y = f(u) 且 u = g(x)，那么 dy/dx = (dy/du) · (du/dx)，即 d/dx [f(g(x))] = f'(g(x)) · g'(x)。

Implicit differentiation uses the chain rule to differentiate both sides of an equation with respect to x, treating y as a function of x.

隐函数求导使用链式法则对方程两边关于x求导，将y视为x的函数。

Logarithmic differentiation: take ln of both sides of y = f(x) when f(x) is a complicated product, quotient, or exponent; then differentiate implicitly.

对数求导法： 当y = f(x)为复杂的积、商或幂指函数时，两边取自然对数，然后隐函数求导。

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

Function	Derivative
sin x	cos x
cos x	-sin x
tan x	sec² x
cot x	-csc² x
sec x	sec x tan x
csc x	-csc x cot x

These derivatives are fundamental and are tested extensively on both AB and BC exams.

这些导数都是基础内容，在AB和BC考试中都会大量考查。

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5. Derivatives of Inverse Trigonometric Functions (BC) | 反三角函数的导数（BC）

For BC students, memorize the derivatives of the inverse trigonometric functions. These are used in integration and series.

BC考生需熟记反三角函数的导数，它们在积分和级数中都会出现。

Function	Derivative
arcsin x (sin⁻¹ x)	1 / √(1 – x²)
arccos x (cos⁻¹ x)	-1 / √(1 – x²)
arctan x (tan⁻¹ x)	1 / (1 + x²)
arccot x (cot⁻¹ x)	-1 / (1 + x²)
arcsec x (sec⁻¹ x)	1 / (|x| √(x² – 1))
arccsc x (csc⁻¹ x)	-1 / (|x| √(x² – 1))

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6. Applications of Derivatives | 导数的应用

Critical points: x-values where f'(x) = 0 or f'(x) does not exist. They are candidates for local extrema.

临界点： 满足 f'(x)=0 或 f'(x) 不存在的x值，它们是局部极值的候选点。

Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that f'(c) = [f(b) – f(a)] / (b – a).

中值定理（MVT）： 若 f 在 [a, b] 上连续且在 (a, b) 上可导，则存在 c ∈ (a, b) 使得 f'(c) = [f(b) – f(a)] / (b – a)。

L’Hôpital’s Rule: For indeterminate forms 0/0 or ∞/∞, lim (x→c) f(x)/g(x) = lim (x→c) f'(x)/g'(x) provided the limit on the right exists.

洛必达法则： 对于 0/0 或 ∞/∞ 未定式，lim (x→c) f(x)/g(x) = lim (x→c) f'(x)/g'(x)，只要右侧极限存在。

lim (x→c) f(x)/g(x) = lim (x→c) f'(x)/g'(x)

Related rates: Differentiating an equation relating two or

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