# 数値微分計算機

> The first or second derivative of any function f(x) at a point, to about 20 significant digits by Richardson extrapolation, with the tangent line drawn.

操作できるページ：https://www.calcopenly.com/ja/math/derivative-calculator
分野：数学の計算機

The derivative f′(x₀) is the slope of f at x₀, the limit of the central difference quotient [f(x₀ + h) − f(x₀ − h)]/2h as h shrinks to 0. The calculator evaluates that quotient for h = 0.1 × max(1, |x₀|) and repeated halvings, then combines the results by Richardson extrapolation (Ridders' method, Numerical Recipes §5.7), which cancels the h², h⁴, … error terms one at a time and reaches about 20 significant digits. The second derivative uses [f(x₀ + h) − 2f(x₀) + f(x₀ − h)]/h² the same way.

Use it to check a derivative worked out by hand, to find a rate of change where no formula is convenient, or to get a tangent line. The default, x² sin x at x = 1, has derivative 2 sin 1 + cos 1 ≈ 2.2232443 and tangent line y = 2.223244x − 1.381773.

Trig functions use radians. When the slopes from the left and right disagree, f has a corner there and the calculator reports that instead of a number.

## 入力

- **Function f(x)**: Use x as the variable, e.g. x^3 - 2x, exp(x), ln(x). Trig functions use radians here.
- **At x =**
- **微分** (選択肢：First f′(x), Second f″(x))
- **Chart half-width**: The chart shows x₀ ± this much

## 結果

- 微分 — 主な結果
- f(x₀)
- Tangent line
- Estimated error

## 計算式

$$
\begin{gathered} f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x-h)}{2h} \\[10pt] f''(x) = \lim_{h \to 0}\frac{\delta^2 f}{h^2} \\[4pt] \delta^2 f = f(x+h) - 2f(x) + f(x-h) \end{gathered}
$$

## 計算例

### d/dx x³ at x = 2

- Function f(x): x^3
- At x =: 2
- 微分: First f′(x)
- **微分: 12**
- **f(x₀): 8**
- **Tangent line: y = 12x − 16**
- 照合元：3x² = 12 at x = 2; tangent 8 + 12(x − 2)

### d/dx sin x at 0 (radians)

- Function f(x): sin(x)
- At x =: 0
- 微分: First f′(x)
- **微分: 1**
- 照合元：cos 0 = 1

### d/dx x²·sin x at 1

- Function f(x): x^2 * sin(x)
- At x =: 1
- 微分: First f′(x)
- **微分: 2.223244275484**
- 照合元：2 sin 1 + cos 1 with sin/cos by Taylor series in Python decimal at 70 digits (hp.py)

### d/dx eˣ at 1

- Function f(x): exp(x)
- At x =: 1
- 微分: First f′(x)
- **微分: 2.718281828459**
- 照合元：e (Python Decimal(1).exp())

### Second derivative of ln x at 2

- Function f(x): ln(x)
- At x =: 2
- 微分: Second f″(x)
- **微分: -0.25**
- 照合元：−1/x² = −1/4 at x = 2

### Flat point: d/dx (x − 1)³ at 1

- Function f(x): (x-1)^3
- At x =: 1
- 微分: First f′(x)
- **微分: 0**
- **f(x₀): 0**
- 照合元：3(x − 1)² = 0 at x = 1

## よくある質問

### What is a derivative?

The derivative of f at x₀ is the slope of its graph there: the limit of [f(x₀ + h) − f(x₀)]/h as h approaches 0. For f(x) = x³ at x = 2 the slope is 3 × 2² = 12, so near x = 2 the function rises about 12 units for each unit of x. The line y = 12x − 16, which touches the curve at (2, 8), is the tangent there.

### How do you find a derivative numerically?

Evaluate a difference quotient with a small step h. The central quotient [f(x + h) − f(x − h)]/2h beats the one-sided [f(x + h) − f(x)]/h because its error shrinks like h² rather than h: for sin x at 0 with h = 0.1 it gives 0.998334 against the exact 1. A tiny h eventually fails through rounding error, so this calculator extrapolates from moderate steps instead.

### What does the second derivative tell you?

The second derivative f″(x) is the rate of change of the slope, so it measures curvature: positive where the graph bends upward, negative where it bends downward. For ln x, f″(x) = −1/x², so f″(2) = −0.25. Numerically it comes from [f(x + h) − 2f(x) + f(x − h)]/h². Where f″ changes sign the graph has an inflection point.

### How do you find the equation of a tangent line?

Use y = f(x₀) + f′(x₀)(x − x₀). For f(x) = x³ at x₀ = 2, f(2) = 8 and f′(2) = 12, so y = 8 + 12(x − 2) = 12x − 16. The tangent is also the best straight-line approximation to f near x₀: it estimates 2.1³ as 8 + 12 × 0.1 = 9.2, against the exact 9.261.

### Why does a function have no derivative at some points?

The slopes from the left and the right must agree. |x| at 0 has slope −1 from the left and +1 from the right, a corner, so it has no derivative there, even though the central quotient averages to 0. A jump, or a vertical tangent such as the cube root of x at 0, also rules one out. The calculator compares one-sided quotients and reports the corner.

### 「数値微分計算機」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：6。 例えば、「d/dx x³ at x = 2」は3x² = 12 at x = 2; tangent 8 + 12(x − 2)と照合しています。

### この計算方法の出典は何ですか？

Press et al., Numerical Recipes (3rd ed.) §5.7 — numerical derivatives (Ridders' method); Wolfram MathWorld — Richardson Extrapolation.

## 出典

- Press et al., Numerical Recipes (3rd ed.) §5.7 — numerical derivatives (Ridders' method)
- [Wolfram MathWorld — Richardson Extrapolation](https://mathworld.wolfram.com/RichardsonExtrapolation.html)