hcju/proofwikips huggingface dataset:
Definition:Generated Algebraic Substructure
Let $\left({A, \circ}\right)$ be an algebraic structure. Let $G \subseteq A$ be any subset of $A$. The '''algebraic substructure generated''' by $G$ is the smallest substructure of $\left({A, \circ}\right)$ which contains $G$. It is written $\left \langle {G} \right \rangle$.
Definition:Generator of Algebra
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$. Let $S \subseteq A_R$ be a subset of $A_R$. The '''subalgebra generated by $S$''' is the smallest subalgebra $B_R$ of $A_R$ which contains $S$.
Definition:Alternative Algebra
Let $\struct {A_R, \oplus}$ be an algebra over a ring $R$. Then $\struct {A_R, \oplus}$ is an alternative algebra  $\oplus$ is an alternative operation. That is: :For all $a, b \in A_R$, the subalgebra generated by $\set {a, b}$ is an associative algebra.
Definition:Power-Associative Algebra
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$. Then $\left({A_R, \oplus}\right)$ is a power-associative algebra  $\oplus$ is power-associative. That is: :For all $a \in A_R$, the subalgebra generated by $\left\{{a}\right\}$ is an associative algebra.
Definition:Cayley-Dickson Construction
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. The '''Cayley-Dickson Construction''' on $A$ is the procedure which generates a new algebra $A'$ from $A$ as follows. Let: : $A' = \left({A'_F, \oplus'}\right) = \left({A, \oplus}\right)^2$ where $\left({A, \oplus}\right)^2$ denotes the Cartesian product of $\left({A, \oplus}\right)$ with itself. Then $\oplus'$ and $*'$ are defined on $A'$ as follows: :$\left({a, b}\right) \oplus' \left({c, d}\right) = \left({a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}\right)$ :${\left({a, b}\right)^*}' = \left({a^*, -b}\right)$ where: : $\left({a, b}\right), \left({c, d}\right) \in A'$ : $a^*$ is the conjugation of $a \in A$.
Definition:Class Membership
To define membership not only for sets, but also for proper classes, we will extend the membership relation to include specific behaviors with proper classes and sets alike: :$\forall A, B: \paren {A \in B \iff \exists x: \paren {A = x \land x \in B } }$ With this definition, no proper classes is a member of any other class, proper or not.
Definition:Universal Class
The '''universal class''' is the class of which all sets are members. The '''universal class''' is defined most commonly in literature as: :$V = \set {x: x = x}$ where $x$ ranges over all sets. It can be briefly defined as the '''class of all sets'''.
Definition:Real Element in Star-Algebra
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Let $a \in A$ be real. Then $\left({a, 0}\right)$ is defined as real in $A'$.  hiap5z48d0c7np6b1ydn6qgk1gnjqd8
Definition:Octonion
The set of '''octonions''', usually denoted $\Bbb O$, can be defined by using the Cayley-Dickson construction from the quaternions $\Bbb H$ as follows: From Quaternions form Algebra, $\Bbb H$ forms a nicely normed $*$-algebra. Let $a, b \in \Bbb H$. Then $\tuple {a, b} \in \Bbb O$, where: :$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$ :$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$ where: :$\overline a$ is the conjugate on $a$ and  :$\overline {\tuple {a, b} }$ is the conjugation operation on $\Bbb O$.
Definition:Quaternion/Construction from Cayley-Dickson Construction
The set of quaternions $\Bbb H$ can be defined by the Cayley-Dickson construction from the set of complex numbers $\C$. From Complex Numbers form Algebra, $\C$ forms a nicely normed $*$-algebra. Let $a, b \in \C$. Then $\left({a, b}\right) \in \Bbb H$, where: :$\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$ :$\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$ where: :$\overline a$ is the complex conjugate of $a$ and  :$\overline {\left({a, b}\right)}$ is the conjugation operation on $\Bbb H$. It is clear by direct comparison with the Construction from Complex Pairs that this construction genuinely does generate the Quaternions.
Definition:Complex Number/Construction from Cayley-Dickson Construction
The complex numbers can be defined by the Cayley-Dickson construction from the set of real numbers $\R$. From Real Numbers form Algebra, $\R$ forms a nicely normed $*$-algebra. Let $a, b \in \R$. Then $\left({a, b}\right) \in \C$, where: :$\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$ :$\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$ where: :$\overline a$ is the conjugate of $a$ and  :$\overline {\left({a, b}\right)}$ is the conjugation operation on $\C$. From Real Numbers form Algebra, $\overline a = a$ and so the above translate into: :$\left({a, b}\right) \left({c, d}\right) = \left({a c - d b, a d + c b}\right)$ :$\overline {\left({a, b}\right)} = \left({a, -b}\right)$ It is clear by direct comparison with the formal definition that this construction genuinely does generate the complex numbers.
Definition:Monomial
=== Monomial of Polynomial Ring === {{:Definition:Monomial of Polynomial Ring}} === Monomial of Free Commutative Monoid === {{:Definition:Monomial of Free Commutative Monoid}}
Definition:Homogeneous Polynomial
A '''homogeneous polynomial''' is a polynomial whose monomials with nonzero coefficients all have the same total degree.
Definition:Quadratic Form
Let $\mathbb K$ be a field of characteristic $\Char {\mathbb K} \ne 2$. Let $V$ be a vector space over $\mathbb K$. A '''quadratic form''' on $V$ is a mapping $q : V \mapsto \mathbb K$ such that: :$\forall v \in V : \forall \kappa \in \mathbb K : \map q {\kappa v} = \kappa^2 \map q v$ :$b: V \times V \to \mathbb K: \tuple {v, w} \mapsto \map q {v + w} - \map q v - \map q w$ is a bilinear form
Definition:Galois Field
A '''Galois field''' $\struct {\GF, +, \circ}$ is a field such that $\GF$ is a finite set.
Definition:Infinite Field
An infinite field $\struct {F, +, \cdot}$ is a field such that $F$ is a infinite set.
Definition:Subtraction/Ring
Let $\struct {R, +, \circ}$ be a ring. The operation of '''subtraction''' $a - b$ on $R$ is defined as: :$\forall a, b \in R: a - b := a + \paren {-b}$ where $-b$ is the (ring) negative of $b$.
Definition:Zero Divisor/Ring
Let $\struct {R, +, \circ}$ be a ring. A '''zero divisor (in $R$)''' is an element $x \in R$ such that either: :$\exists y \in R^*: x \circ y = 0_R$ or: :$\exists y \in R^*: y \circ x = 0_R$ where $R^*$ is defined as $R \setminus \set {0_R}$. That is, such that $x$ is either a left zero divisor or a right zero divisor. The expression: : '''$x$ is a zero divisor''' can be written: : $x \divides 0_R$
Definition:Zero Divisor/Algebra
Let $\struct {A_R, \oplus}$ be an algebra over a ring $\struct {R, +, \cdot}$. Let the zero vector of $A_R$ be $\mathbf 0_R$. Let $a, b \in A_R$ such that $a \ne \mathbf 0_R$ and $b \ne \mathbf 0_R$. Then $a$ and $b$ are '''zero divisors of $A_R$''' : :$a \oplus b = \mathbf 0_R$
Definition:Multiplicative Group
Let $\struct {F, +, \times}$ be a field. Let $F^* := F \setminus \set 0$ be the set $F$ less its zero. The group $\struct {F^*, \times}$ is known as the '''multiplicative group of $F$'''.
Definition:Multiplicative Identity
Let $\struct {F, +, \times}$ be a field. Then the identity element of the multiplicative group $\struct {F^*, \times}$ of $F$ is called the multiplicative identity of $F$. It is often denoted $e_F$ or $1_F$, or, if there is no danger of ambiguity, $e$ or $1$. {{refactor|Either merge this concept with that of Definition:Unity of Field or enter this statement as a page in its own right.}} Note that the multiplicative identity of $F$ is the unity of the ring that $\struct {F, +, \times}$ is by definition of a field.
Definition:Multiplicative Inverse/Field
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ such that $a \ne 0_F$. Then the inverse element of $a$ with respect to the $\times$ operator is called the '''multiplicative inverse''' of $F$. It is usually denoted $a^{-1}$ or $\dfrac 1 a$.
Definition:Affine Transformation
Let $\mathcal E$ and $\mathcal F$ be affine spaces with difference spaces $E$ and $F$ respectively. Let $\mathcal L: \mathcal E \to \mathcal F$ be a mapping. Then $\mathcal L$ is an '''affine transformation''' or '''affine mapping''' if there exists a linear transformation $L: E \to F$ such that for every pair of points $p, q \in \mathcal E$: :$\map {\mathcal L} q = \map {\mathcal L} p + \map L {\vec {p q} }$
Definition:Annihilator of Ring
Let $B: R \times \Z$ be a bilinear mapping defined as: :$B: R \times \Z: \tuple {r, n} \mapsto n \cdot r$ where $n \cdot r$ defined as an integral multiple of $r$: :$n \cdot r = r + r + \cdots \paren n \cdots r$ Note the change of order of $r$ and $n$: :$\map B {r, n} = n \cdot r$ Let $D \subseteq R$ be a subring of $R$. Then the '''annihilator''' of $D$ is defined as: :$\map {\mathrm {Ann} } D = \set {n \in \Z: \forall d \in D: n \cdot d = 0_R}$ or, when $D = R$: :$\map {\mathrm {Ann} } R = \set {n \in \Z: \forall r \in R: n \cdot r = 0_R}$ It is seen to be, therefore, the set of all integers whose integral multiples, with respect to the elements of a ring or a field, are all equal to the zero of that ring or field. === Trivial Annihilator === {{:Definition:Trivial Annihilator}}
Definition:Ring Endomorphism
Let $\struct {R, +, \circ}$ be a ring. Let $\phi: R \to R$ be a (ring) homomorphism from $R$ to itself. Then $\phi$ is a ring endomorphism.
Definition:Zero Homomorphism
Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively. Consider the mapping $\zeta: R_1 \to R_2$ defined as: :$\forall r \in R_1: \map \zeta r = 0_2$ Then $\zeta$ is '''the zero homomorphism from $R_1$ to $R_2$'''.
Definition:Identity Automorphism
Let $\struct {R, +, \circ}$ be a ring. Consider the identity mapping $I_R: R \to R$: :$\forall r \in R: \map {I_R} r = r$ Then $I_R$ is '''the identity automorphism on $R$'''.
Definition:Field Monomorphism
Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields. Let $\phi: F \to K$ be a (field) homomorphism. Then $\phi$ is a field monomorphism  $\phi$ is an injection.
Definition:Isomorphism (Abstract Algebra)/F-Isomorphism
Let $R, S$ be rings with unity. Let $F$ be a subfield of both $R$ and $S$. Let $\phi: R \to S$ be an $F$-homomorphism such that $\phi$ is bijective. Then $\phi$ is an '''$F$-isomorphism'''. The relationship between $R$ and $S$ is denoted $R \cong_F S$.
Definition:R-Algebraic Structure Epimorphism
Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures. Then $\phi: S \to T$ is an '''$R$-algebraic structure epimorphism''' : : $(1): \quad \phi$ is a surjection : $(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$ : $(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$ This definition also applies to modules, and also to vector spaces.
Definition:Field Epimorphism
Let $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ be fields. Let $\phi: R \to S$ be a (field) homomorphism. Then $\phi$ is a field epimorphism  $\phi$ is a surjection.
Definition:Group Endomorphism
Let $\struct {G, \circ}$ be a group. Let $\phi: G \to G$ be a (group) homomorphism from $G$ to itself. Then $\phi$ is a group endomorphism.
Definition:R-Algebraic Structure Endomorphism
Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ be an $R$-algebraic structure. Let $\phi: S \to S$ be an $R$-algebraic structure homomorphism from $S$ to itself. Then $\phi$ is an $R$-algebraic structure endomorphism. This definition continues to apply when $S$ is a module, and also when it is a vector space.
Definition:Field Endomorphism
Let $\struct {F, +, \circ}$ be a field. Let $\phi: F \to F$ be a (field) homomorphism from $F$ to itself. Then $\phi$ is a field endomorphism.
Definition:Homogeneous Linear Equations
A '''system of homogeneous linear equations''' is a set of simultaneous linear equations: :$\displaystyle \forall i \in \closedint 1 m: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ such that all the $\beta_i$ are equal to zero: :$\displaystyle \forall i \in \closedint 1 m : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ That is: {{begin-eqn}} {{eqn | l = 0       | r = \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n }} {{eqn | l = 0       | r = \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n }} {{eqn | o = \cdots}} {{eqn | l = 0       | r = \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n }} {{end-eqn}} === Matrix Representation === {{:Definition:Homogeneous Linear Equations/Matrix Representation}}
Definition:Set/Explicit Set Definition
A (finite) set can be defined by '''explicitly''' specifying ''all'' of its elements between the famous curly brackets, known as '''set braces''': $\set {}$. When a set is defined like this, note that ''all'' and ''only'' the elements in it are listed. This is called '''explicit (set) definition'''. It is possible for a set to contain other sets. For example: :$S = \set {a, \set a}$
Definition:Set/Implicit Set Definition
If the elements in a set have an ''obvious'' pattern to them, we can define the set '''implicitly''' by using an ellipsis ($\ldots$). For example, suppose $S = \set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$. A more compact way of defining this set is: :$S = \set {1, 2, \ldots, 10}$ With this notation we are asked to suppose that the numbers count up uniformly, and we can read this definition as: :'''$S$ is the set containing $1$, $2$, and so on, up to $10$.'''
Definition:Set/Definition by Predicate
An object can be specified by means of a '''predicate''', that is, in terms of a property (or properties) that it possesses. Whether an object $x$ possesses a particular (characteristic) property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\map P x$. Hence a set can be specified by means of such a propositional function: :$S = \set {x: \map P x}$ which means: :'''$S$ is the set of all objects which have the property $P$''' or, more formally: :'''$S$ is the set of all $x$ such that $\map P x$ is true.''' In this context, we see that the symbol $:$ is interpreted as '''such that'''.
Definition:Real Number/Operations on Real Numbers
We interpret the following symbols: {{begin-axiom}} {{axiom | n = \text R 1         | lc= Negative         | q = \forall a \in \R         | m = \exists ! \paren {-a} \in \R: a + \paren {-a} = 0 }} {{axiom | n = \text R 2         | lc= Minus         | q = \forall a, b \in \R         | m = a - b = a + \paren {-b} }} {{axiom | n = \text R 3         | lc= Reciprocal         | q = \forall a \in \R \setminus \set 0         | m = \exists ! a^{-1} \in \R: a \times \paren {a^{-1} } = 1 = \paren {a^{-1} } \times a         | rc= it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ }} {{axiom | n = \text R 4         | lc= Divided by         | q = \forall a \in \R \setminus \set 0         | m = a \div b = \dfrac a b = a / b = a \times \paren {b^{-1} }         | rc= it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ }} {{end-axiom}} The validity of all these operations is justified by Real Numbers form Field.
Definition:Real Number/Real Number Line
From the Cantor-Dedekind Hypothesis, the set of real numbers is isomorphic to any infinite straight line. The '''real number line''' is an arbitrary infinite straight line each of whose points is identified with a real number such that the distance between any two real numbers is consistent with the length of the line between those two points. :800px
Definition:Bounded Above Set
Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''bounded above (in $S$)'''  $T$ admits an upper bound (in $S$).
Definition:Bounded Above Mapping
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is '''bounded above on $S$''' by the upper bound $H$ : :$\forall x \in S: \map f x \preceq H$ That is,  $f \sqbrk S = \set {\map f x: x \in S}$ is bounded above by $H$.
Definition:Supremum of Set
Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. An element $c \in S$ is the '''supremum of $T$ in $S$''' : :$(1): \quad c$ is an upper bound of $T$ in $S$ :$(2): \quad c \preceq d$ for all upper bounds $d$ of $T$ in $S$. If there exists a '''supremum''' of $T$ (in $S$), we say that: :'''$T$ admits a supremum (in $S$)''' or :'''$T$ has a supremum (in $S$)'''.
Definition:Supremum of Mapping/Real-Valued Function
{{:Definition:Supremum of Mapping/Real-Valued Function/Definition 1}}
Definition:Maximal/Ordered Set
{{:Definition:Maximal/Ordered Set/Definition 1}}
Definition:Maximum Value of Real Function/Absolute
Let $f: \R \to \R$ be a real function. Let $f$ be bounded above by a supremum $B$. It may or may not be the case that $\exists x \in \R: \map f x = B$. If such a value exists, it is called the '''maximum''' of $f$ on $S$, and that this '''maximum''' is '''attained at $x$'''.
Definition:Bounded Below Set
Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''bounded below (in $S$)'''  $T$ admits a lower bound (in $S$).
Definition:Bounded Below Mapping
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is said to be '''bounded below (in $T$)''' by the lower bound $L$ : :$\forall x \in S: L \preceq \map f x$ That is, iff $f \sqbrk S = \set {\map f x: x \in S}$ is bounded below by $L$.
Definition:Upper Bound of Set
Let $\struct {S, \preceq}$ be an ordered set. Let $T$ be a subset of $S$. An '''upper bound for $T$ (in $S$)''' is an element $M \in S$ such that: :$\forall t \in T: t \preceq M$ That is, $M$ succeeds every element of $T$.
Definition:Upper Bound of Mapping
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Let $f$ be bounded above in $T$ by $H \in T$. Then $H$ is an '''upper bound of $f$'''.
Definition:Lower Bound of Set
Let $\struct {S, \preceq}$ be an ordered set. Let $T$ be a subset of $S$. A '''lower bound for $T$ (in $S$)''' is an element $m \in S$ such that: :$\forall t \in T: m \preceq t$ That is, $m$ precedes every element of $T$.
Definition:Lower Bound of Mapping
Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Let $f$ be bounded below in $T$ by $H \in T$. Then $H$ is a '''lower bound of $f$'''.
Definition:Infimum of Set
Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. An element $c \in S$ is the '''infimum of $T$ in $S$''' : :$(1): \quad c$ is a lower bound of $T$ in $S$ :$(2): \quad d \preceq c$ for all lower bounds $d$ of $T$ in $S$. If there exists an '''infimum''' of $T$ (in $S$), we say that '''$T$ admits an infimum (in $S$)'''.
Definition:Infimum of Mapping
Let $S$ be a set. Let $\struct {T, \preceq}$ be an ordered set. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f \sqbrk S$, the image of $f$, admit an infimum. Then the '''infimum''' of $f$ (on $S$) is defined by: :$\displaystyle \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$ === Real-Valued Function === {{:Definition:Infimum of Mapping/Real-Valued Function}}
Definition:Bounded Ordered Set
Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be both bounded below and bounded above in $S$. Then $T$ is '''bounded in $S$'''.
Definition:Bounded Mapping
Let $\struct {T, \preceq}$ be an ordered set. Let $f: S \to T$ be a mapping. Let the image of $f$ be bounded. Then $f$ is '''bounded'''. That is, $f$ is '''bounded'''  it is both bounded above and bounded below.
Definition:Minimal/Ordered Set
{{:Definition:Minimal/Ordered Set/Definition 1}}
Definition:Minimum Value of Real Function/Absolute
Let $f: \R \to \R$ be a real function. Let $f$ be bounded below by an infimum $B$. It may or may not be the case that $\exists x \in \R: \map f x = B$. If such a value exists, it is called the '''minimum value''' of $f$ on $S$, and this '''minimum''' is '''attained at $x$'''.
Definition:Minimal/Set
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\mathcal T \subseteq \powerset S$ be a subset of $\powerset S$. Let $\struct {\mathcal T, \subseteq}$ be the ordered set formed on $\mathcal T$ by $\subseteq$ considered as an ordering. Then $T \in \mathcal T$ is a '''minimal set''' of $\mathcal T$  $T$ is a minimal element of $\struct {\mathcal T, \subseteq}$. That is: :$\forall X \in \mathcal T: X \subseteq T \implies X = T$
Definition:Continuous Real Function/Interval
=== Open Interval === This is a straightforward application of continuity on a set. {{Definition:Continuous Real Function/Open Interval}} === Closed Interval === {{Definition:Continuous Real Function/Closed Interval}} === Half Open Intervals === Similar definitions apply to half open intervals: {{Definition:Continuous Real Function/Half Open Interval}}
Definition:Derivative/Complex Function
The definition for a complex function is similar to that for real functions. === At a Point === {{:Definition:Derivative/Complex Function/Point}} === On an Open Set === {{:Definition:Derivative/Complex Function/Open Set}}
Definition:Derivative/Real Function/Derivative at Point
Let $I$ be an open real interval. Let $f: I \to \R$ be a real function defined on $I$. Let $\xi \in I$ be a point in $I$. Let $f$ be differentiable at the point $\xi$. ==== Definition 1 ==== {{:Definition:Derivative/Real Function/Derivative at Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Derivative/Real Function/Derivative at Point/Definition 2}}
Definition:Derivative/Real Function/Derivative on Interval
Let $I\subset\R$ be an open interval. Let $f : I \to \R$ be a real function. Let $f$ be differentiable on the interval $I$. Then the '''derivative of $f$''' is the real function  $f': I \to \R$ whose value at each point $x \in I$ is the derivative $f' \left({x}\right)$: :$\displaystyle \forall x \in I: f' \left({x}\right) := \lim_{h \mathop \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h$
Definition:Derivative/Real Function
=== At a Point === {{:Definition:Derivative/Real Function/Derivative at Point}} === On an Open Interval === {{:Definition:Derivative/Real Function/Derivative on Interval}} === With Respect To === {{:Definition:Derivative/Real Function/With Respect To}}
Definition:Derivative/Higher Derivatives/Order of Derivative
The '''order''' of a derivative is the '''number of times it has been differentiated'''. For example: :a first derivative is of '''first order''', or '''order $1$''' :a second derivative is of '''second order''', or '''order $2$''' and so on.
Definition:Derivative/Higher Derivatives
=== Second Derivative === {{:Definition:Derivative/Higher Derivatives/Second Derivative}} === Third Derivative === {{:Definition:Derivative/Higher Derivatives/Third Derivative}} === Higher Order Derivatives === Higher order derivatives are defined in similar ways: {{:Definition:Derivative/Higher Derivatives/Higher Order}} === First Derivative === If derivatives of various orders are being discussed, then what has been described here as the derivative is frequently referred to as the '''first derivative''': {{:Definition:Derivative/Real Function/Derivative on Interval}} == Order of Derivative == {{:Definition:Derivative/Higher Derivatives/Order of Derivative}}