Δ 差商计算器（任意多项式，化简并求极限）Difference Quotient Calculator (Any Polynomial, Simplified with the Limit)

使用说明与常见问题How-to & FAQ

「差商计算器（任意多项式，化简并求极限）」是什么？What is the Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) tool?

输入多项式 f(x) 的系数（从最高次到常数项，如 1, 0, -3, 2 表示 x³ − 3x + 2），把差商 [f(x + h) − f(x)] / h 展开化简成关于 x 和 h 的式子，给出 h → 0 时的极限（导数 f′(x)），并可代入具体的 x 和 h 求值。本地处理。Enter the coefficients of a polynomial f(x) from the highest power to the constant (for example 1, 0, -3, 2 for x³ − 3x + 2) to expand and simplify the difference quotient [f(x + h) − f(x)] / h in x and h, get its limit as h → 0 (the derivative f′(x)), and evaluate it at a chosen x and h. Processed locally.

如何使用「差商计算器（任意多项式，化简并求极限）」？How do I use the Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) tool?

在「差商计算器（任意多项式，化简并求极限）」中，填写「系数（最高次 → 常数）」、「x 值（可选）」、「h 值（可选）」等参数，结果会随输入实时计算并显示在下方。In the Difference Quotient Calculator (Any Polynomial, Simplified with the Limit): Fill in "Coefficients (highest power to constant)", "x value (optional)", "h value (optional)"; results are recalculated instantly as you type.

差商是什么？What is the difference quotient?

[f(x + h) − f(x)] / h 是函数在 x 到 x + h 之间的平均变化率，也就是割线的斜率。h 趋于 0 时它的极限就是导数 f′(x)。[f(x + h) − f(x)] / h is the average rate of change of f between x and x + h, the slope of the secant line. Its limit as h goes to 0 is the derivative f′(x).

怎么化简？How is it simplified?

把 f(x + h) 用二项式定理展开，减去 f(x) 后所有不含 h 的项都会抵消，剩下的每一项都含 h，可以约去分母的 h。例如 f(x) = x² 时差商为 2x + h。Expand f(x + h) with the binomial theorem; after subtracting f(x), every term without h cancels and the rest share a factor h, which cancels with the denominator. For f(x) = x² the quotient is 2x + h.

系数怎么输入？How do I enter the coefficients?

从最高次项到常数项依次输入，缺少的次数填 0。例如 2x³ − x + 5 输入 2, 0, -1, 5。List them from the highest power down to the constant, with 0 for any missing power. For 2x³ − x + 5 enter 2, 0, -1, 5.

能举个计算实例吗？Can you show a worked example?

例如在「差商计算器（任意多项式，化简并求极限）」中填 系数（最高次 → 常数） = 1, 0, 0，x 值（可选） = 2，h 值（可选） = 0.001，结果区显示：f(x)：x²；差商 [f(x+h) − f(x)]/h：2x + h；h → 0 的极限 f′(x)：2x；代入 x = 2, h = 0.001：4.001；f′(2)：4For example, entering Coefficients (highest power to constant) = 1, 0, 0, x value (optional) = 2, h value (optional) = 0.001 in the Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) gives: f(x): x²; Difference quotient [f(x+h) − f(x)]/h: 2x + h; Limit as h → 0, f′(x): 2x; At x = 2, h = 0.001: 4.001; f′(2): 4

哪些输入会被检查？Which inputs are validated?

「差商计算器（任意多项式，化简并求极限）」会检查这些输入——「x 值（可选）」、「h 值（可选）」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) checks its inputs — "x value (optional)", "h value (optional)": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「差商计算器（任意多项式，化简并求极限）」的设置：「系数（最高次 → 常数）」默认 1, 0, 0；「x 值（可选）」默认 2；「h 值（可选）」默认 0.001。每次生成结果都按当前设置处理。Settings in the Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) — "Coefficients (highest power to constant)" default 1, 0, 0; "x value (optional)" default 2; "h value (optional)" default 0.001. The result always uses the current settings.

计算精度如何？How precise are the results?

「差商计算器（任意多项式，化简并求极限）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「差商计算器（任意多项式，化简并求极限）」免费吗？需要注册吗？Is the Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) free? Do I need an account?

「差商计算器（任意多项式，化简并求极限）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Difference Quotient Calculator (Any Polynomial, Simplified with the Limit) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.