Derivatives of common elementary functions table
Table shows derivatives of selected elementary functions of one variable f(x) such as linear function, square function, sine, cosine, logarithm etc.

# 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.