式を入力
f(x) = は不要です手順
適用した順に並べた各規則答え
Second and higher-order derivatives
Differentiate again to get acceleration, concavity, and the test that classifies turning points.
Differentiating twice gives the second derivative, written \(f''(x)\), \(\frac{d^{2}y}{dx^{2}}\) or \(y''\). It is the rate at which the rate is changing: if \(f\) is position and \(f'\) is velocity, then \(f''\) is acceleration. Set the Order control on the calculator above to 2 — or use the smaller one here — and the working shows every round.
What the second derivative tells you
- \(f'' > 0\): the curve is concave up, holding water, and \(f'\) is increasing.
- \(f'' < 0\): the curve is concave down, and \(f'\) is decreasing.
- \(f'' = 0\) with a sign change on either side: an inflection point, where the curvature flips.
The second derivative test
At a point where \(f'(a) = 0\), the second derivative classifies it without checking signs on both sides: \(f''(a) > 0\) means a minimum, \(f''(a) < 0\) means a maximum, and \(f''(a) = 0\) is inconclusive — fall back to the first derivative test there.
Third, fourth and nth derivatives
Keep going and the notation switches from primes to \(f^{(4)}\), \(f^{(5)}\), \(f^{(n)}\). Some functions settle into a pattern: \(e^{x}\) is unchanged at every order, \(\sin x\) cycles back to itself every four, and a polynomial of degree \(n\) reaches zero at order \(n+1\). Others grow without ever repeating — a fifth-order derivative of a product can run to a hundred steps, which is why the working is written out a batch at a time.
Derivative Calculator の仕組み
-
解析
入力した文字列が演算の木に変換され、省略された掛け算の記号は自動で補われます。
-
組版
木を数式表記で表示するので、計算機が意図どおりに読み取ったかを確認できます。
-
微分
ノードごとに規則を一つ — 和、積、商、べき乗、連鎖律 — それぞれが手順として記録されます。
-
簡約と作図
結果は厳密な有理数演算で簡約され、両方の曲線が標本化されて描かれます。
記法と規則
パーサーが受け付ける記法と、エンジンが適用する規則の一覧です。規則名はそれぞれの解説ページに繋がっています。
適用される規則
| 規則 | 式 |
|---|---|
| Constant | \(\frac{d}{dx}c = 0\) |
| Variable | \(\frac{d}{dx}x = 1\) |
| Sum | \(\left(f + g\right)' = f' + g'\) |
| Constant multiple | \(\left(cf\right)' = c\,f'\) |
| Power | \(\frac{d}{dx}x^{n} = n x^{n-1}\) |
| Product | \(\left(fg\right)' = f'g + fg'\) |
| Quotient | \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^{2}}\) |
| Chain | \(\left(f \circ g\right)' = f'(g)\,g'\) |
| Exponential | \(\frac{d}{dx}a^{x} = a^{x}\ln a\) |
| Logarithm | \(\frac{d}{dx}\ln x = \frac{1}{x}\) |
| Logarithm, base a | \(\frac{d}{dx}\log_{a} x = \frac{1}{x\ln a}\) |
| Trigonometric | \(\frac{d}{dx}\sin x = \cos x\) |
| Inverse trig | \(\frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}\) |
| Hyperbolic | \(\frac{d}{dx}\tanh x = \operatorname{sech}^{2} x\) |
| Inverse hyperbolic | \(\frac{d}{dx}\operatorname{artanh} x = \frac{1}{1 - x^{2}}\) |
| Absolute value | \(\frac{d}{dx}\left|x\right| = \operatorname{sgn} x, \; x \neq 0\) |
| Logarithmic | \(\frac{d}{dx}f^{g} = f^{g}\left(g'\ln f + \frac{gf'}{f}\right)\) |
| Rewrite in natural logs | \(\log_{a} b = \frac{\ln b}{\ln a}\) |
入力の記法
| 入力 | 意味 |
|---|---|
2x, 3(x+1) | Implicit multiplication |
x^3, x**3 | Powers — x² and x³ also work |
sqrt(x), √x | Square root |
cbrt(x) | Cube root |
e^x, exp(x) | Exponential function |
ln(x), log(x) | Natural logarithm |
log(2, x), log2(x) | Logarithm to a given base |
sin^2(x) | The square of sin x |
tan^-1(x), arctan(x) | Inverse tangent |
|x-1|, abs(x-1) | Absolute value |
pi, e, tau | Constants |
x_1, theta | Subscripted and Greek variables |
More derivative tools
The calculator above handles ordinary derivatives to any order. These pages take the same engine into the cases that need their own controls or their own explanation.
Partial derivatives
Work with two or three variables and differentiate with respect to any one of them.
Derivative at a point
Get the slope as a number at any x, and the equation of the tangent line there.
Derivative graphs
Plot a function against its derivative and read the turning points straight off.
Implicit differentiation
Find dy/dx for curves such as x² + y² = 25 that are not solved for y.
Parametric and polar
Curves given by x(t), y(t) or by r(θ), where dy/dx comes from the parameter.
Directional derivatives
The gradient of a surface, and how steeply it rises in any direction you choose.
Numerical differentiation
Estimate a derivative from values alone, and compare it with the exact answer.
Derivative of an integral
The Fundamental Theorem, and the chain rule for a variable limit.
Learn the mathematics
Explanations rather than tools, for when the answer is not the point.
What is a derivative?
The definition, the notation, and where derivatives sit in each calculus course.
First principles
The limit definition worked slowly, for x², √x and sin x.
Practice problems
Problems grouped by rule, with answers and the standard derivatives table.
Formula cheat sheet
Every rule and standard derivative on one printable page.
Looking for a specific rule? 規則 lists all eighteen, each with worked examples and a graph.
よくある質問
すべての手順が表示されますか。
はい。適用した規則はすべて順番に記録され、使った一般公式と短い説明が添えられます。長い式では手順をまとめて少しずつ表示し、ページの速度を保ちます。
どの関数に対応していますか。
多項式と根号、任意の底の指数関数と対数関数、六つの三角関数とその逆関数、双曲線関数とその逆関数、さらに絶対値と符号関数に対応しています。
高階の導関数も求められますか。
はい、十階まで求められます。各回の微分は完全に計算され、一回目の手順が詳しく表示されます。
入力した内容はサーバーに送られますか。
いいえ。解析、微分、簡約、作図はすべてブラウザ内で行われます。最近の入力はお使いの端末のローカルストレージにのみ保存されます。
スマートフォンでも使えますか。
はい。レイアウトは一列に折り返し、グラフはドラッグと二本指ズームに対応します。関数キーボードから、携帯では入力しにくい記号を呼び出せます。