식 입력
f(x) = 는 생략하세요풀이 단계
적용한 순서대로 나열한 규칙답
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 작동 방식
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구문 분석
입력한 문자열이 연산 트리로 바뀌고, 생략된 곱셈 기호는 자동으로 채워집니다.
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조판
트리를 수식 표기로 보여 주므로 계산기가 의도한 대로 읽었는지 확인할 수 있습니다.
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미분
노드마다 하나의 규칙 — 합, 곱, 몫, 거듭제곱, 연쇄 — 이 적용되고 각각 풀이 단계로 기록됩니다.
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정리와 그래프
결과를 정확한 유리수 연산으로 정리한 뒤 두 곡선을 표본화해 그립니다.
표기법과 규칙
파서가 받아들이는 표기와 엔진이 적용하는 규칙입니다. 규칙 이름은 각각의 설명 페이지로 연결됩니다.
적용 규칙
| 규칙 | 식 |
|---|---|
| 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 | \(\log_{a} b = \frac{\ln b}{\ln a}\) |
입력 표기법
| 입력 | 의미 |
|---|---|
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? 규칙 lists all eighteen, each with worked examples and a graph.
자주 묻는 질문
모든 풀이 단계를 보여 주나요?
네. 엔진이 적용한 규칙은 순서대로 기록되며, 사용한 일반 공식과 짧은 설명이 함께 표시됩니다. 긴 식은 단계를 나누어 보여 주어 페이지가 빠르게 유지됩니다.
어떤 함수를 지원하나요?
다항식과 근호, 임의의 밑을 가진 지수·로그 함수, 여섯 가지 삼각함수와 그 역함수, 쌍곡선 함수와 그 역함수, 절댓값과 부호 함수를 지원합니다.
고차 도함수도 구할 수 있나요?
네, 10차까지 가능합니다. 각 차수를 모두 계산하며 첫 번째 차수의 풀이 과정을 자세히 보여 줍니다.
입력한 내용이 서버로 전송되나요?
아니요. 구문 분석, 미분, 정리, 그래프 그리기가 모두 브라우저에서 이루어집니다. 최근 입력은 사용자의 기기에만 저장됩니다.
휴대전화에서도 쓸 수 있나요?
네. 화면이 한 열로 정리되고, 그래프는 끌기와 두 손가락 확대를 지원하며, 함수 키보드가 휴대전화로 입력하기 번거로운 기호를 제공합니다.