Gustavo Radbruch

The derived from a function is the means of determining how much this function varies when the quantity of which it depends, its argument, changes. More precisely, a derivative is an expression (numerical or algebraic) giving the relationship between the infinitesimal variations of the function and the infinitesimal variations of its argument. For example, speed is the derivative of displacement compared to time, and acceleration is the derivative, compared to time, speed.

The concept of derivative was born at the 17th century in the writings of Leibniz and Newton which names it Fluxion and which defines it as “the ultimate quotient of two increases évanescents”.

The derivative of the function $f\; \backslash ,$ is noted in mathematics $f\text{'}\; \backslash ,$. The physicists prefer the notation $\backslash \; frac$ which with the advantage of pointing out the name of the argument, but one meets also the notation $\backslash \; dowry\; f$ which is in the case of used a derivative compared to time, noted $t$. In the particular case of a partial Derivative, the notation $\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}$ is used.

The concept of derivative is a basic concept in analyzes functional. It makes it possible to study the variations of a function, to build tangents with a curve and to solve problems of optimization.

Intuitive approach

To approach this concept in an intuitive way, let us start with us to give a Courbe representative of a function in a Cartesian reference mark, continues, i.e. traced of only one feature of pencil, and well “ smooth ”, one will say there that the associated function is derivable. Whatever the point which one chooses on the curve, one will be able to then trace a tangent what is called, i.e. a line which locally marries the direction of this curve. Concretely, if one plots the curve and his tangent and that one approaches by zoomant sufficient, one has more and more evil to distinguish the curve from his tangent. One includes/understands easily that if the curve “goes up” ( IE if the associated function is increasing), the tangent will be also rising; conversely, if the function is decreasing, the tangent will be downward. If one gives oneself a X-coordinate $x\; \backslash ,\; \_0$ for which the function $f\; \backslash ,$ is derivable, one calls number derived from $f\; \backslash ,$ in $x\; \backslash ,\; \_0$ the directing coefficient of the tangent to the curve at the point of X-coordinate $x\; \backslash ,\; \_0$. This real gives invaluable information on the local behavior of a function: it is the algebraic measurement of the Speed to which this function changes when its Variable changes. For a function with several variables, one speaks about the Dérivée partial compared to one of his variables.

Thus, if the number derived from a function is positive on an interval, this function will be increasing on this same interval. Conversely, if it is negative, it will be decreasing. When the derived number is null in a point, the curve admits a horizontal tangent in this point.

In the example opposite:

• in 0, the curve goes down, therefore the number derived there is negative (it is worth -1)
• in 1, the curve always goes down, but the slope is less there (- 0,5).
• in 2, the curve is perfectly horizontal, therefore the derivative is null (0).
• in 3, the curve goes up, therefore the number derived there is positive (0,5).

Approaches historical

See also: History of the infinitesimal calculus

As of second half of the 17th century, the mathematical field of the numerical analysis knew an extraordinary projection thanks to work of Newton and Leibniz as regards differential and integral calculus , in particular treating concept of infinitely small and his relationship with the sums known as integral . It is however Blaise Pascal which, in first half of the XVII|17 e century, has the first led of the studies on the tangent concept of to a curve - itself called them “ touching ”; the marquis of Hospital will also take part at the end of the XVII|17 e to pack this new theory, in particular by using the derivative to calculate a limit in the case of particular unspecified forms (cf Rule of the Hospital). Wallis, mathematician English (especially known for the continuation of integrals which bears its name) also contributed to the rise of the differential analysis.

Nevertheless this theory just hatched is not equipped yet with all the mathematical rigor which it would have required, and in particular concept of infinitely small introduced by Newton, which holds more the intuitive one, and which could generate errors since one does not get along of course what is or not Négligeable. It is at the 18th century that of Alembert introduces the more rigorous definition of the number derived as a limit from the rate of increase - in a form similar to that which is used and taught nowadays. However, at the time of Alembert, it is the concept of limit which this time poses problem: $\backslash \; R$ is not built yet formally (see on this subject the article on the real numbers and the article detailed on their construction) and it is only with work of Weierstraß in the middle of the 19th century that the concept of derivative will be entirely formalized.

It is in the passing to Lagrange (fine of the XVIII|18 e century) that one owes the notation $f\text{'}\; \backslash \; left\; (X\; \backslash \; right)$, today completely usual, to indicate the number derived from $f\; \backslash ,$ in $x\; \backslash ,$.

Formal definition

That is to say $f\; \backslash ,$ a real function with actual values definite on an unspecified meeting of Interval S noncommonplace, and $x\; \backslash ,\; \_0$ pertaining to the interior of the whole of definition $\backslash \; mathcal\; \{D\}\; \_f$.

For all $h\; \backslash \; in\; \backslash \; R^*$ such as $\backslash \; sub\; \backslash \; mathcal\; \{D\}\; \_f$, one calls rate of increase in $f\; \backslash ,$ in $x\; \backslash ,\; \_0$ and with a step of $h\; \backslash ,$ the quantity:

$t\_\; \{x\_0\}\; (H)\; =\; \{F\; (x\_0+h)\; -\; F\; (x\_0)\; \backslash \; over\; H\}$

It is about the directing coefficient of the right-hand side connecting the points of coordinates $(x\_0,\; F\; (x\_0))$ and $(x\_0+h,\; F\; (x\_0+h))$. If $t\_\; \{x\_0\}\; (H)$ admits a finished limit when $h\; \backslash ,$ tends towards 0, one says that $f$ is derivable in $x\; \backslash ,\; \_0$, in which case the number derived from $f\; \backslash ,$ in $x\; \backslash ,\; \_0$ is equal to the limit of this rate of increase. One notes then:

$f\text{'}\; (x\_0)\; =\; \backslash \; lim\_\; \{H\; \backslash \; to\; 0\}\; t\_\; \{x\_0\}\; (H)\; =\; \backslash \; lim\_\; \{H\; \backslash \; to\; 0\}\; \{F\; (x\_0+h)\; -\; F\; (x\_0)\; \backslash \; over\; H\}$

Or, in an equivalent way:

$f\text{'}\; (x\_0)\; =\; \backslash \; lim\_\; \{X\; \backslash \; to\; x\_0\}\; \{F\; (X)\; -\; F\; (x\_0)\; \backslash \; over\; x-x\_0\}$

Function for which the rate of increase in a point admits a limit (which is the derived number) is known as derivable in this point.

Derivation can as be defined for functions of a real variable with values on other wholes as $\backslash \; R$.

For example, a function $f\; \backslash ,$ of a real variable, with values in $\backslash \; R^n$, are derivable in $x\; \backslash ,\; \_0$ if and only if all its Coordonnée S is derivable in $x\; \backslash ,\; \_0$; and its derivative is the function whose coordinates are the derivative of the coordinates of $f\; \backslash ,$. It is a particular case of functions of vectorial variable and with value in a vector Space normalized or metric.

Derived function

The Dérivabilité is a priori a local concept (derivability in a point), but if a function is derivable in any point of an interval, one can define his derived function on the interval in question. The function derived from $f$, noted $f\text{'}\; \backslash ,$ (to pronounce “F precedes” ) - or $\backslash \; frac$, even $\backslash \; dowry\; \{F\}$ (in physical sciences or industrial), is defined on $\backslash \; mathfrak\; \{D\}\; \_f$ and the field of derivability of $f\; \backslash ;$ (together of the points of $\backslash \; R$ in which $f$ is derivable) is defined by:

$f\text{'}:\backslash ,\; \backslash \; mathfrak\; \{D\}\; \_f\; \backslash \; rightarrow\; \backslash \; R,\; \backslash \; X\; \backslash \; mapsto\; f\text{'}\; (X)$

It is the function which takes in any point of $\backslash \; mathfrak\; \{D\}\; \_f$ the value of the number derived from $f\; \backslash ,$ in this point.

Thus, when the derivable function $f$ is increasing, the derived function $f\text{'}\; \backslash ,$ is positive. $f\text{'}\; \backslash ,$ is cancelled at the points where $f$ admits tangent horizontal.

The derived functions are used in particular in the study of the real functions and the differential equations. A function which is equal to its derivative is known as Exponentielle (this one is solution of $y\text{'}=y$, cf detailed article).

Notations

There exist various notations to express the derivative of a function. One distinguishes:

• the notation of Lagrange:

$f\text{'}\; \backslash \; left\; (X\; \backslash \; right)$

• the notation of Leibniz:

$\backslash \; frac$ which is equivalent, more rigorously, with $\backslash \; frac$

• the notation of Isaac Newton:

$\backslash \; dowry\; \{X\}\; =\; \backslash \; frac\; =\; x\text{'}\; (T)$ which is rather used in Physique.

• Lastly, the notation of Euler:

$D\_x\; F\; (X)\; \backslash ;$

Derived from the usual functions

See also: Derivative usual

Rules of derivation

See also: Operations on the derivative

$f\text{'}\; \backslash ,$ can be often calculated directly starting from an expression of $f\; \backslash ,$, when it is about a “simple” function, by using the tables of the usual Dérivées. For functions which are expressed like linear combination of simple functions, like produces, quotient or made up, one uses a small number of algebraic rules deduced from the definition given higher. The rules most usually used are the following ones:

In particular, here current rules resulting from derived from made up:

Let us specify that in this last table, the made up one of the functions $g$ and $f$ was not noted $g\; \backslash \; circ\; f$ as it would be appropriate but $g\; (F)$. These notations can lend to confusion but are quite convenient to retain the formulas and to apply them.

Demonstrations

Theorem of the finished increases

See also: Theorem of the finished increases

; Statement

If a function ƒ is continuous on an interval and derivable on] has , B then it exists a point '' X '' ₀ of has , B such as the derivative in this point is the rate of variation between '' has '' and '' B '' : $f\text{'}\; (x\_0)\; =\; \backslash \; frac\; \{F\; (B)\; -\; F\; (a)\}\; \{Ba\}$

; Demonstration

One starts by showing this when ƒ ( has ) = ƒ ( B ): the function is not strictly monotonous, therefore either the function is a horizontal line and the property is valid in any point, or the function presents a maximum or a minimum and thus its derivative are cancelled in a point.

For an unspecified function, one is reduced to the preceding case by withdrawing the function

$g\; (X)\; =\; \backslash \; frac\; \{F\; (B)\; -\; F\; (a)\}\; \{Ba\}\; \backslash \; cdot\; (x-a)$

then (ƒ- G ) ( B ) = (ƒ- G ) ( has ) = ƒ ( has ) and one applies the linearity of the derivative.

This property is used in Cinématique to determine an approximation of the vector Speed starting from a raised of point.

Derivation of reciprocal of a function

That is to say $\backslash \; mathit\; \{F\}\; \backslash ,$ a derivable and strictly monotonous function of the interval $\backslash \; mathcal\; \{I\}$ on the interval $\backslash \; mathcal\; \{J\}\; =\; \{F\; (\backslash \; mathcal\; \{I\})\}$ and if $f\text{'}\; \backslash ,$ is not cancelled by on $\backslash \; mathcal\; \{I\}$, then the function $f^\; \{-\; 1\}\; \backslash ,$ is derivable on $\backslash \; mathcal\; \{J\}$ and:

$(f^\; \{-\; 1\})\; “=\; \backslash \; frac\; \{1\}\; \{F”\; \backslash \; circ\; f^\; \{-\; 1\}\}$

Demonstration:

Let us show that $f^\; \{-\; 1\}\; \backslash ,\; \backslash !$ derivable in $B\; =\; F\; (a)\; \backslash ,\; \backslash !$. By supposing $f\text{'}\; (A)\; \backslash \; 0\; \backslash ,\; \backslash !$, let us show that $\backslash \; lim\_\; \{there\; \backslash \; to\; B\}\; \{\backslash \; frac\; \{f^\; \{-\; 1\}\; (there)\; -\; f^\; \{-\; 1\}\; (b)\}\; \{there\; -\; B\}\}\; =\; \{1\; \backslash \; over\; \{f\text{'}\; (A)\}\}\; \backslash ,\; \backslash !$.

The function $f\; \backslash ,\; \backslash !$ being derivable in $a\; \backslash ,\; \backslash !$, one a:

$f\text{'}\; (A)\; =\; \backslash \; lim\_\; \{X\; \backslash \; to\; has\}\; \{\backslash \; frac\; \{F\; (X)\; -\; F\; (a)\}\; \{x-a\}\}\; \backslash ,\; \backslash !$

How $f^\; \{-\; 1\}\; \backslash ,\; \backslash !$ is continuous in $b\; \backslash ,\; \backslash !$, the theorem of composition of the limits gives:

$f\text{'}\; (A)\; =\; \backslash \; lim\_\; \{there\; \backslash \; to\; B\}\; \{\backslash \; frac\; \{F\; (f^\; \{-\; 1\}\; (there))\; -\; F\; (a)\}\; \{f^\; \{-\; 1\}\; (there)\; -\}\}\; =\; \backslash \; lim\_\; \{has\; there\; \backslash \; to\; B\}\; \{\backslash \; frac\; \{y-b\}\; \{f^\; \{-\; 1\}\; (there)\; -\; f^\; \{-\; 1\}\; (b)\}\}$

This limit being nonnull, according to the theorem on the reverse of a limit, one a:

$\backslash \; lim\_\; \{there\; \backslash \; to\; B\}\; \{\backslash \; frac\; \{f^\; \{-\; 1\}\; (there)\; -\; f^\; \{-\; 1\}\; (b)\}\; \{y-b\}\}\; =\; \{1\; \backslash \; over\; f\text{'}\; (A)\}$

Or: $(f^\; \{-\; 1\})\; “(b)\; =\; \{1\; \backslash \; over\; \{F”\; (f^\; \{-\; 1\}\; (b))\}\; \}$

Derived from reciprocal from the goniometrical functions

One uses the preceding result to establish:

• If $f\; (X)\; =\; \backslash \; operatorname\; \{Arcsin\}\; (X)\; \backslash ,$, then $f\text{'}\; (X)\; =\; \{1\; \backslash \; over\; \backslash \; sqrt\; \{1-x^2\}\}$

• If $f\; (X)\; =\; \backslash \; operatorname\; \{Arcsin\}\; (\backslash \; varphi\; (X))\backslash ,$, then $f\text{'}\; (X)\; =\; \{\backslash \; varphi\text{'}\; (X)\; \backslash \; over\; \{\backslash \; sqrt\; \{1\; \backslash \; varphi\; (X)\; ^2\}\}\}$
• If $f\; (X)\; =\; \backslash \; operatorname\; \{Arccos\}\; (X)\; \backslash ,$, then $f\text{'}\; (X)\; =\; \{-\; 1\; \backslash \; over\; \backslash \; sqrt\; \{1-x^2\}\}$
• If $f\; (X)\; =\; \backslash \; operatorname\; \{Arccos\}\; (\backslash \; varphi\; (X))\backslash ,$, then $f\text{'}\; (X)\; =\; \{-\; \backslash \; varphi\text{'}\; (X)\; \backslash \; over\; \{\backslash \; sqrt\; \{1\; \backslash \; varphi\; (X)\; ^2\}\}\}$
• If $f\; (X)\; =\; \backslash \; operatorname\; \{Arctan\}\; (X)\; \backslash ,$, then $f\text{'}\; (X)\; =\; \{1\; \backslash \; over\; \{1+x^2\}\}$
• If $f\; (X)\; =\; \backslash \; operatorname\; \{Arctan\}\; (\backslash \; varphi\; (X))\backslash ,$, then $f\text{'}\; (X)\; =\; \{\backslash \; varphi\text{'}\; (X)\; \backslash \; over\; \{1+\; \backslash \; varphi\; (X)\; ^2\}\}$

$\backslash \; varphi$ indicates a derivable function with value in the field of derivability of the function considered.

Derived from order N

See also: reiterated Derivation

One defines the derivative of order $n$ for a derivable function $n$ time by recurrence:

$\backslash frac=\backslash frac\; \backslash frac$

$\backslash \; frac$ is also noted $f^\; \{(N)\}\backslash ,$.

Formulate of Leibniz

If $f,\; g$ is derivable functions $n$ time, then:

$(fg)\; ^\; \{(N)\}=\; \backslash \; sum\_\; \{k=0\}\; ^\; \{N\}\; \{N\; \backslash \; choose\; K\}\; f^\; \{(K)\}g^\; \{(n-k)\}$.

In particular for $n=2$,

$(fg)$ =f g+2f' g'+fg \, \! .

One will note the analogy with the Formule of the binomial theorem.

Notation of Leibniz

See also: Notation of Leibniz

Derived from the dependant rates of variation

Analyzes of a derived function

By finding the values of X where the derivative is worth 0 or does not exist, one finds the critical numbers of the function. The critical numbers of F make it possible to implicitly find its maxima and its minima. To carry out the Test of the derived first, one builds a Tableau of variation; if the sign of the derived function passes from at least in front of a number criticizes, there is a maximum and if the sign of the derived function passes at least to more in front of the number criticizes, there is a minimum. Moreover, when the sign of derived first is positive, the function goes up; if it is negative, it goes down. One concludes nothing, so at the point criticizes the function does not change a sign. By deriving the derivative first, one with the Derived second. To carry out the Test of the derived second, one finds the numbers critical of the derived first to place them in the same table; when one observes a change of sign of the derived second in front of this or these critical numbers, one says that one has one (or of) not (S) of inflection. The points of inflection mark a change of the concavity of the function. A positive sign of the derived second means that the function is concave to the top and a negative sign of the derived second means that the function is concave downwards. Knowing the changes of concavity and the extrema of the function, one can then trace a draft of the graph.

Derived and optimization

Method to optimize an output using differential calculus:

1. Mathématisation

2. * Definitions and drawing: the unknown variables are defined and one represents them on a diagram.
3. * To write the Function objective with two variables and to specify if one seeks a maximum or a minimum in the situation given.
4. * To find the relation between the two Variable S.
5. * To write the Function objective with a variable and to specify the Field function.
6. Analysis
7. * To derive the function to obtain the Derived first.
8. * To find the numbers critical of the function, where the derivative first is worth zero or does not exist in the intervals of the field.
9. * To carry out the Test of the derived first or the Test of the derived second to determine the maximum or the required minimum of the situation.
10. One formulates the answer in a concise way compared to the question.

Derived and asymptotic

See also: Asymptotic

Once one determined the asymptotes of the function, one can note them in the Tableau of variation to trace the draft of the graph adequately.

See too

• Examples of calculation of derived
• reiterated Derivation
• Theorem from the finished increases
• Theorem in Darboux (analyzes)

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