İfadeyi girin
f(x) = yazmanıza gerek yokAdımlar
her kural uygulanma sırasıylaCevap
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 nasıl çalışır
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Ayrıştırma
Yazdığınız metin bir işlem ağacına dönüşür; atlanan çarpım işaretleri kendiliğinden eklenir.
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Dizgi
Ağaç matematiksel gösterimde çizilir, böylece hesaplayıcının kastettiğinizi okuduğunu doğrulayabilirsiniz.
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Türev alma
Her düğümde tek bir kural — toplam, çarpım, bölüm, kuvvet, zincir — ve her biri bir adım olarak kaydedilir.
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Sadeleştirme ve grafik
Sonuç tam aritmetikle sadeleştirilir, ardından iki eğri de örneklenip çizilir.
Gösterim ve kurallar
Ayrıştırıcının kabul ettiği her şey ve motorun uyguladığı kurallar. Her kural adı kendi sayfasına gider.
Uygulanan kurallar
| Kural | İfade |
|---|---|
| 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 | \(\log_{a} b = \frac{\ln b}{\ln a}\) |
Girdi söz dizimi
| Yazdığınız | Anlamı |
|---|---|
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? Kurallar lists all eighteen, each with worked examples and a graph.
Sorular
Hesaplayıcı her adımı gösteriyor mu?
Evet. Motorun uyguladığı her kural sırayla kaydedilir; kullanılan genel bağıntı ve kısa bir açıklama da eklenir. Uzun ifadelerin adımları sayfa hızlı kalsın diye küme küme yazılır.
Hangi fonksiyonlar destekleniyor?
Polinomlar ve kökler, her tabanda üstel ve logaritmik fonksiyonlar, altı trigonometrik fonksiyon ve tersleri, hiperbolik aile ve tersleri, ayrıca mutlak değer ve işaret fonksiyonu.
Yüksek mertebeden türev alabiliyor mu?
Evet, onuncu mertebeye kadar. Her tur tam olarak hesaplanır ve ayrıntılı adımlar ilk tur için gösterilir.
Girdiğim ifade sunucuya gönderiliyor mu?
Hayır. Ayrıştırma, türev alma, sadeleştirme ve grafik çizimi tarayıcınızda olur. Son ifadeler yalnızca kendi cihazınızda saklanır.
Telefonda kullanabilir miyim?
Evet. Yerleşim tek sütuna iner, grafik sürükleme ve iki parmakla yakınlaştırmayı kabul eder, fonksiyon klavyesi de telefonda yazması zor simgeleri sunar.