Usual Derivatives — Robin Pourtaud

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

Differentiability domain	Function	Derivative of the Function	Example
$\mathbb{R}$	$k$	$0$	$f'(10) = 0$
$\mathbb{R}$	$kx$	$k$	$f'(5x) = 5$
$\mathbb{R}$	$x^n$ ( $n\in\mathbb{N}^*$ )	$nx^{n-1}$	$f'(x^{10}) = 10x^9$
$\mathbb{R}^{*+}$	$\ln(x)$	$\frac{1}{x}$	$f'(\ln(x)) = \frac{1}{x}$
$\mathbb{R}$	$e^x$	$e^x$	$f'(e^x) = e^x$
$\mathbb{R}$	$\sin(x)$	$\cos(x)$	$f'(\sin(x)) = \cos(x)$
$\mathbb{R}$	$\cos(x)$	$-\sin(x)$	$f'(\cos(x)) = -\sin(x)$
$\mathbb{R}\setminus\{\frac{(2k+1)\pi}{2}, k\in\mathbb{Z}\}$	$\tan(x)$	$1 + \tan^2(x)$	$f'(\tan(x)) = 1 + \tan^2(x)$
$\mathbb{R}\setminus\{k\pi, k\in\mathbb{Z}\}$	$\cot(x)$	$-1 - \cot^2(x)$	$f'(\cot(x)) = -1 - \cot^2(x)$
$\mathbb{R}^{*}$	$x^n$ (negative)	$nx^{n-1}$	$f'(x^{-3}) = -3x^{-4}$
$(0, +\infty)$	$\sqrt{x}$	$\frac{1}{2\sqrt{x}}$	$f'(\sqrt{x}) = \frac{1}{2\sqrt{x}}$
$\mathbb{R}^*$ for odd $n$ , $(0,+\infty)$ for even $n$	$\sqrt[n]{x}$ ( $n\geq2$ )	$\frac{1}{n(\sqrt[n]{x})^{n-1}}$	$f'(\sqrt[3]{x}) = \frac{1}{3(\sqrt[3]{x})^2}$
$\mathbb{R}$	$a^x$ (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.