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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.
Bagaimana Derivative Calculator berfungsi
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Menghurai
Teks anda menjadi pokok operasi, dan tanda darab yang ditinggalkan dimasukkan secara automatik.
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Menatur
Pokok itu dipaparkan dalam tatatanda matematik supaya anda boleh menyemak bahawa kalkulator membaca apa yang anda maksudkan.
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Membezakan
Satu hukum bagi setiap nod — hasil tambah, hasil darab, hasil bahagi, kuasa, rantai — setiap satu direkodkan sebagai langkah.
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Memudahkan dan melukis
Keputusan dipermudah dengan aritmetik tepat, kemudian kedua-dua lengkung dilukis.
Tatatanda dan hukum
Semua yang diterima penghurai, dan hukum yang digunakan oleh enjin. Setiap nama hukum menuju ke halamannya sendiri.
Hukum yang digunakan
| Hukum | Pernyataan |
|---|---|
| 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 | \(\log_{a} b = \frac{\ln b}{\ln a}\) |
Sintaks input
| Anda taip | Maksudnya |
|---|---|
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? Hukum lists all eighteen, each with worked examples and a graph.
Soalan lazim
Adakah kalkulator menunjukkan setiap langkah?
Ya. Setiap hukum yang digunakan direkodkan mengikut urutan, bersama hukum am yang dipakai dan penjelasan ringkas. Ungkapan panjang dipaparkan sekumpulan pada satu masa supaya halaman kekal pantas.
Fungsi apa yang disokong?
Polinomial dan punca, eksponen dan logaritma pada sebarang asas, keenam-enam fungsi trigonometri dan songsangannya, keluarga hiperbolik dan songsangannya, serta nilai mutlak dan fungsi tanda.
Bolehkah ia mengira terbitan peringkat tinggi?
Boleh, sehingga peringkat kesepuluh. Setiap pusingan dibezakan sepenuhnya dan langkah terperinci ditunjukkan bagi pusingan pertama.
Adakah input saya dihantar ke pelayan?
Tidak. Penghuraian, pembezaan, pemudahan dan lukisan graf semuanya berlaku dalam pelayar anda. Ungkapan terkini disimpan hanya pada peranti anda.
Bolehkah saya menggunakannya pada telefon?
Boleh. Susun atur bertukar kepada satu lajur, graf menerima seretan dan zum dua jari, dan papan kekunci fungsi menyediakan simbol yang menyusahkan untuk ditaip pada telefon.