Derivative formulas

Function f(x)	Derivative f'(x)	Notice
$a$	$0$
$x$	$1$
$ax+b$	$a$
$ax^2+bx+c$	$2ax+b$
$x^a$	$ax^{a-1}$
$\sqrt{x}$	$\frac{1}{2\sqrt{x}}$
$\sqrt[n]{x}$	$\frac{1}{n\sqrt[n]{x^{n-1}}}$	$n \in N \backslash \{0,1\}$
$\frac{1}{x}$	$\frac{-1}{x^2}$
$\frac{a}{x}$	$\frac{-a}{x^2}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tg x$	$\frac{1}{\cos^2 x}$
$\ctg x$	$-\frac{1}{\sin^2 x}$
$a^x$	$a^x \cdot \ln a$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$
$\ln\|x\|$	$\frac{1}{x}$
$\log_ax$	$\frac{1}{x \ln a}$
$arc \sin x$	$\frac{1}{\sqrt{1 - x^2}}$
$arc \cos x$	$\frac{-1}{\sqrt{1 - x^2}}$
$arc \tg x$	$\frac{1}{1 + x^2}$
$arc \ctg x$	$\frac{-1}{1 + x^2}$

Some facts

The derivative in point is defined as the limit of so-called difference quotient with x going to this point:

$f'(x_0) = \lim\limits_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$
Above formula is often given in alternative form (which is sometimes more useful depending on context). In this form we assume $x = x_0 + h$ , where $h$ is so-called "very small change" of x variable (independent variable):

$f'(x) = \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}$
Derivative understood as a function, assigns the above formula (i.e. limit of difference quotient) to each point from domain.

$f': x \rightarrow \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}$
ⓘ Example: The derivative of linear function $f(x) = ax + b$ , is f'(x) = a, because:
$(ax+b)' \overset{\mathrm{def}}{=} \lim\limits_{h \to 0} \frac{a \cdot (x + h) + b - (ax + b)}{h} = \lim\limits_{h \to 0} \frac{\cancel{ax} + ah + \cancel{b} - \cancel{ax} - \cancel{b}}{h} = \lim\limits_{h \to 0} \frac{a\cancel{h}}{\cancel{h}} = a$
Derivate of function f(x) is often marked with the symbol f'(x) (read: "f prim from x").
In practice, derivatives are rarely calculated directly from definition. For everyday usage, ready-made derivatives are used.
Sometimes it is difficult or impossible to calculate derivative in the analytical form (i.e. given by explicite formula such as -sinx). In this case, the numerical calculations may be alternative. They are based on calculating the derivative value on the selected range using the definition of the difference quotient directly, assuming a finite, but "contractually small" increment value h, e.g. 0.00001. In practice, the value of the increment is selected experimentally for a specific application. In this way you can get approximate values of the derivative, which is often used in natural or engineering sciences.
The unique and, due to this fact, very interesting function for mathematicians, is $f(x) = e^x$ . It's derivate is the same as original function.

$(e^x)' = e^x$
The derivative can be interpreted as the measure of variability of the function. Such interpretation is particularly useful in natural and engineering sciences, e.g.
- in physics velocity is a derivative of position over time, so it's the quantity determining how quickly the position of the body changes over time,
- in electronics, the current intensity is defined as a derivative of the flow of electric charge over time,
- in chemistry dipole moment is a derivative of the molecule energy over the intensity of the electric field, in other words it tells how much the external electric field will affect the energy of the molecule,
- etc.

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