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This article has been written by Robin Pourtaud ([email protected]) and published on December 1, 2023.
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Reminders
- The domain of definition of a function is the set of values for which the function is defined. For example, the function $f(x) = \sqrt{x}$ is defined for $x \geq 0$.
- The derivative of a function at a point gives the instantaneous rate of change of the function at that point. It is fundamental in analysis for studying the variations of functions.
Table of Usual Derivatives
| Domain of Definition | Function | Derivative of the Function | Example |
|---|---|---|---|
| $\mathbb{R}$ | $k$ | $0$ | $f’(10) = 0$ |
| $\mathbb{R}$ | $kx$ | $k$ | $f’(5x) = 5$ |
| $\mathbb{R}$ | $x^k$ | $kx^{k-1}$ | $f’(x^{10}) = 10x^9$ |
| $\mathbb{R}^{*+}$ | $\ln(x)$ | $\frac{1}{x}$ | $f’(\ln(x)) = \frac{1}{x}$ |
| $\mathbb{R}$ | $e^x$ | $e^x$ | $f’(e^x) = e^x$ |
| $\mathbb{R}$ | $\sin(x)$ | $\cos(x)$ | $f’(\sin(x)) = \cos(x)$ |
| $\mathbb{R}$ | $\cos(x)$ | $-\sin(x)$ | $f’(\cos(x)) = -\sin(x)$ |
| $\mathbb{R}\setminus{k\pi, k\in\mathbb{Z}}$ | $\tan(x)$ | $1 + \tan^2(x)$ | $f’(\tan(x)) = 1 + \tan^2(x)$ |
| $\mathbb{R}\setminus{\frac{(2k+1)\pi}{2}, k\in\mathbb{Z}}$ | $\cot(x)$ | $-1 - \cot^2(x)$ | $f’(\cot(x)) = -1 - \cot^2(x)$ |
| $\mathbb{R}^{*}$ | $x^n$ (negative) | $nx^{n-1}$ | $f’(x^{-3}) = -3x^{-4}$ |
| $(0, +\infty)$ | $\sqrt{x}$ | $\frac{1}{2\sqrt{x}}$ | $f’(\sqrt{x}) = \frac{1}{2\sqrt{x}}$ |
| $\mathbb{R}^{*+}$ | $x^{\frac{1}{n}}$ (n positive integer) | $\frac{1}{n}x^{\frac{1}{n}-1}$ | $f’(x^{\frac{1}{3}}) = \frac{1}{3}x^{-\frac{2}{3}}$ |
| $\mathbb{R}$ | $a^x$ (a > 0, a ≠ 1) | $a^x\ln(a)$ | $f’(2^x) = 2^x\ln(2)$ |
| $\mathbb{R}$ | $\sinh(x)$ | $\cosh(x)$ | $f’(\sinh(x)) = \cosh(x)$ |
| $\mathbb{R}$ | $\cosh(x)$ | $\sinh(x)$ | $f’(\cosh(x)) = \sinh(x)$ |
| $\mathbb{R}$ | $\tanh(x)$ | $1 - \tanh^2(x)$ | $f’(\tanh(x)) = 1 - \tanh^2(x)$ |
This extended table offers a more complete view of common derivatives in mathematics, covering a wide range of functions, including some exponential and hyperbolic functions. These derivatives are essential for various applications in analysis and physics.