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

Function f(x) | Derivative f'(x) | Notice |

$a$ | $0$ | $$ |

$x$ | $1$ | $$ |

$ax+b$ | $a$ | $$ |

$ax^2+bx+c$ | $2ax+b$ | $$ |

$x^a$ | $ax^{a-1}$ | $$ |

$\sqrt{x}$ | $\frac{1}{2\sqrt{x}}$ | $$ |

$\sqrt[n]{x}$ | $\frac{1}{n\sqrt[n]{x^{n-1}}}$ | $n \in N \backslash \{0,1\}$ |

$\frac{1}{x}$ | $\frac{-1}{x^2}$ | $$ |

$\frac{a}{x}$ | $\frac{-a}{x^2}$ | $$ |

$\sin x$ | $\cos x$ | $$ |

$\cos x$ | $-\sin x$ | $$ |

$\tg x$ | $\frac{1}{\cos^2 x}$ | $$ |

$\ctg x$ | $-\frac{1}{\sin^2 x}$ | $$ |

$a^x$ | $a^x \cdot \ln a$ | $$ |

$e^x$ | $e^x$ | $$ |

$\ln x$ | $\frac{1}{x}$ | $$ |

$\ln|x|$ | $\frac{1}{x}$ | $$ |

$\log_ax$ | $\frac{1}{x \ln a}$ | $$ |

$arc \sin x$ | $\frac{1}{\sqrt{1 - x^2}}$ | $$ |

$arc \cos x$ | $\frac{-1}{\sqrt{1 - x^2}}$ | $$ |

$arc \tg x$ | $\frac{1}{1 + x^2}$ | $$ |

$arc \ctg x$ | $\frac{-1}{1 + x^2}$ | $$ |

**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.

- in physics

Tags:

derivative_formulas · function_derivatives · derivatives_of_one_variable_functions · derivatives_of_elementary_functions · derivatives_of_trigonometric_functions · derivatives_of_polynomials · derivative_of_linear_function · derivative_of_square_function · derivative_of_e_to_x · exponents_derivative · logarithm_derivative · natural_logarithm_derivative

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