avewright/proofwiki-math huggingface dataset:
Theorem: Derivative of Hyperbolic Cotangent

:$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x = \dfrac {-1} {\sinh^2 x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\coth x}
      | r = \map {\dfrac \d {\d x} } {\frac {\cosh x} {\sinh x} }
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\sinh x \dfrac \d {\d x} \cosh x - \cosh x \dfrac \d {\d x} \sinh x} {\sinh^2 x}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = \frac {\sinh x \sinh x - \cosh x \dfrac \d {\d x} \cosh x} {\sinh^2 x}
      | c = Derivative of Hyperbolic Cosine
}}
{{eqn | r = \frac {\sinh x \sinh x - \cosh x \cosh x} {\sinh^2 x}
      | c = Derivative of Hyperbolic Sine
}}
{{eqn | r = \frac {-1} {\sinh^2 x}
      | c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = -\csch^2 x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Secant

:$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech x}
      | r = \map {\frac \d {\d x} } {\frac 1 {\cosh x} }
      | c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \map {\frac \d {\d x} } {\paren {\cosh x}^{-1} }
      | c = Exponent Laws
}}
{{eqn | r = -\paren {\cosh x}^{-2} \sinh x
      | c = Derivative of Hyperbolic Cosine, Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \frac {-1} {\cosh x} \frac {\sinh x} {\cosh x}
      | c = Exponent Combination Laws
}}
{{eqn | r = -\sech z \tanh z
      | c = {{Defof|Hyperbolic Secant}} and {{Defof|Hyperbolic Tangent}}
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Secant

:$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$

Proof:
{{begin-eqn}}
{{eqn	| l = \map {\frac \d {\d x} } {\sech x}
	| r = 2 \map {\frac \d {\d x} } {\frac {e^x} {e^{2 x} + 1} }
	| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn	| r = \frac 2 {\paren {e^{2 x} + 1}^2} \paren {\map {\frac \d {\d x} } {e^x} \paren {e^{2 x} + 1} - e^x \map {\frac \d {\d x} } {e^{2 x} + 1} }
	| c = Quotient Rule for Derivatives
}}
{{eqn	| r = -\frac 2 {\paren {e^{2 x} + 1}^2} \paren {2 e^{2 x} \cdot e^x - e^x \cdot e^{2 x} - e^x}
	| c = Derivative of Exponential Function
}}
{{eqn	| r = -\frac {2 \paren {e^{3 x} - e^x} } {\paren {e^{2 x} + 1}^2}
}}
{{eqn	| r = -\frac {2 e^x} {\paren {e^{2 x} + 1} } \cdot \frac {e^{2 x} - 1} {e^{2 x} + 1} 
}}
{{eqn	| r = -\sech x \tanh x
	| c = {{Defof|Hyperbolic Secant}}, {{Defof|Hyperbolic Tangent}}
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Cosecant

:$\map {\dfrac \d {\d x} } {\csch x} = -\csch x \coth x$

Proof:
It is noted that at $x = 0$, $\csch x$ is undefined.
Hence the restriction of the domain.
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\csch x}
      | r = \map {\dfrac \d {\d x} } {\frac 1 {\sinh x} }
      | c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\paren {\sinh z}^{-1} }
      | c = Exponent Laws
}}
{{eqn | r = -\paren {\sinh x}^{-2} \cosh x
      | c = Derivative of Hyperbolic Cosine, Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \frac {-1} {\sinh x} \ \frac {\cosh x} {\sinh x}
      | c = Exponent Combination Laws
}}
{{eqn | r = -\csch x \coth x
      | c = {{Defof|Hyperbolic Cosecant}} and {{Defof|Hyperbolic Cotangent}}
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Inverse Hyperbolic Sine Function

:$\map {\dfrac \d {\d x} } {\arsinh u} = \dfrac 1 {\sqrt {1 + u^2} } \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arsinh u}
      | r = \map {\frac \d {\d u} } {\arsinh u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\sqrt {1 + u^2} } \frac {\d u} {\d x}
      | c = Derivative of Inverse Hyperbolic Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Real Area Hyperbolic Cosine of Function

:$\map {\dfrac \d {\d x} } {\arcosh u} = \dfrac 1 {\sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$ 
where $u > 1$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcosh u}
      | r = \map {\frac \d {\d u} } {\arcosh u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\sqrt {u^2 - 1} } \frac {\d u} {\d x}
      | c = Derivative of Real Area Hyperbolic Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Inverse Hyperbolic Tangent Function

:$\map {\dfrac \d {\d x} } {\artanh u} = \dfrac 1 {1 - u^2} \dfrac {\d u} {\d x}$ 
where $\size u < 1$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\artanh u}
      | r = \map {\frac \d {\d u} } {\artanh u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {1 - u^2} \frac {\d u} {\d x}
      | c = Derivative of Inverse Hyperbolic Tangent
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Inverse Hyperbolic Cotangent Function

:$\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac {-1} {u^2 - 1} \dfrac {\d u} {\d x}$ 
where $\size u > 1$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\coth^{-1} u}
      | r = \map {\frac \d {\d u} } {\coth^{-1} u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {-1} {u^2 - 1} \frac {\d u} {\d x}
      | c = Derivative of Inverse Hyperbolic Cotangent
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Inverse Hyperbolic Secant Function

:$\map {\dfrac \d {\d x} } {\sech^{-1} u} = \dfrac {-1} {u \sqrt {1 - u^2} } \dfrac {\d u} {\d x}$ 
where $0 < u < 1$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech^{-1} u}
      | r = \map {\frac \d {\d u} } {\sech^{-1} u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {-1} {u \sqrt {1 - u^2} } \frac {\d u} {\d x}
      | c = Derivative of Inverse Hyperbolic Secant
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Inverse Hyperbolic Cosecant Function

:$\map {\dfrac \d {\d x} } {\arcsch u} = \dfrac {-1} {\size u \sqrt {1 + u^2} } \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsch u}
      | r = \map {\frac \d {\d u} } {\arcsch u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {-1} {\size u \sqrt {1 + u^2} } \frac {\d u} {\d x}
      | c = Derivative of Inverse Hyperbolic Cosecant
}}
{{end-eqn}}
{{qed}}
Theorem: Intersection of Closed Sets is Closed/Topology

Let $T = \struct {S, \tau}$ be a topological space.
Then the intersection of an arbitrary number of closed sets of $T$ (either finitely or infinitely many) is itself closed.

Proof:
Let $I$ be an indexing set (either finite or infinite).
Let $\ds \bigcap_{i \mathop \in I} V_i$ be the intersection of a indexed family of closed sets of $T$ indexed by $I$.
Then from De Morgan's laws: Difference with Intersection:
:$\ds S \setminus \bigcap_{i \mathop \in I} V_i = \bigcup_{i \mathop \in I} \paren {S \setminus V_i}$
By definition of closed set, each of $S \setminus V_i$ are by definition open in $T$.
We have that $\ds \bigcup_{i \mathop \in I} \paren {S \setminus V_i}$ is the union of a indexed family of open sets of $T$ indexed by $I$.
Therefore, by definition of a topology:
:$\ds \bigcup_{i \mathop \in I} \paren {S \setminus V_i} = S \setminus \bigcap_{i \mathop \in I} V_i$ 
is likewise open in $T$.
Then by definition of closed set, $\ds \bigcap_{i \mathop \in I} V_i$ is closed in $T$.
{{qed}}
Theorem: Finite Union of Closed Sets is Closed/Topology

Let $T = \struct {S, \tau}$ be a topological space.
Then the union of finitely many closed sets of $T$ is itself closed.

Proof:
Let $\ds \bigcup_{i \mathop = 1}^n V_i$ be the union of a finite number of closed sets of $T$.
Then from De Morgan's laws:
:$\ds S \setminus \bigcup_{i \mathop = 1}^n V_i = \bigcap_{i \mathop = 1}^n \paren {S \setminus V_i}$
By definition of closed set, each of the $S \setminus V_i$ is by definition open in $T$.
We have that $\ds \bigcap_{i \mathop = 1}^n \paren {S \setminus V_i}$ is the intersection of a finite number of open sets of $T$.
Therefore, by definition of a topology, $\ds \bigcap_{i \mathop = 1}^n \paren {S \setminus V_i} = S \setminus \bigcup_{i \mathop = 1}^n V_i$ is likewise open in $T$.
Then by definition of closed set, $\ds \bigcup_{i \mathop = 1}^n V_i$ is closed in $T$.
{{qed}}
Theorem: Closed Ball is Closed/Metric Space

Let $M = \struct {A, d}$ be a metric space.
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$.
Then $\map {B_\epsilon^-} x$ is a closed set of $M$.

Proof:
We show that the complement $A \setminus B_\epsilon^- \left({x}\right)$ is open in $M$.
Let $a \in A \setminus \map {B_\epsilon^-} x$.
Then by definition of closed ball:
:$\map d {x, a} > \epsilon$
Put:
:$\delta := \map d {x, a} - \epsilon > 0$
Then:
:$\map d {x, a} - \delta = \epsilon$
Let $b \in \map {B_\delta} a$.
Then:
{{begin-eqn}}
{{eqn | l = \map d {x, b}
      | o = \ge 
      | r = \map d {x, a} - \map d {a, b}
      | c = Reverse Triangle Inequality
}}
{{eqn | o = >
      | r = \map d {x, a} - \delta
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
and so:
:$b \notin \map {B_\epsilon^-} x$
Then:
:$\map {B_\delta} a \subseteq A \setminus \map {B_\epsilon^-} x$
so $A \setminus \map {B_\epsilon^-} x$ is open in $M$.
Hence, by definition of closed set:
:$\map {B_\epsilon^-} x$ is closed in $M$.
{{qed}}
Category:Closed Sets (Metric Spaces)
Category:Closed Ball is Closed
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Theorem: Derivative of Sine of Function

:$\map {\dfrac \d {\d x} } {\sin u} = \cos u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sin u}
      | r = \map {\frac \d {\d u} } {\sin u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \cos u \frac {\d u} {\d x}
      | c = Derivative of Sine Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Cosine of Function

:$\map {\dfrac \d {\d x} } {\cos u} = -\sin u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos u}
      | r = \map {\frac \d {\d u} } {\cos u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = -\sin u \frac {\d u} {\d x}
      | c = Derivative of Cosine Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Tangent of Function

:$\map {\dfrac \d {\d x} } {\tan u} = \sec^2 u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\tan u}
      | r = \map {\frac \d {\d u} } {\tan u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \sec^2 u \frac {\d u} {\d x}
      | c = Derivative of Tangent Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Cotangent of Function

:$\map {\dfrac \d {\d x} } {\cot u} = -\csc^2 u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cot u}
      | r = \map {\frac \d {\d u} } {\cot u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = -\csc^2 u \frac {\d u} {\d x}
      | c = Derivative of Cotangent Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Secant of Function

:$\map {\dfrac \d {\d x} } {\sec u} = \sec u \tan u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sec u}
      | r = \map {\frac \d {\d u} } {\sec u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \sec u \tan u \frac {\d u} {\d x}
      | c = Derivative of Secant Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Cosecant of Function

:$\map {\dfrac \d {\d x} } {\csc u} = \csc u \cot u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\csc u}
      | r = \map {\frac \d {\d u} } {\csc u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = -\csc u \cot u \frac {\d u} {\d x}
      | c = Derivative of Cosecant Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of General Logarithm of Function

:$\map {\dfrac \d {\d x} } {\log_a u} = \dfrac {\log_a e} u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\log_a u}
      | r = \map {\frac \d {\d u} } {\log_a u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {\log_a e} u \frac {\d u} {\d x}
      | c = Derivative of General Logarithm Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Natural Logarithm of Function

:$\map {\dfrac \d {\d x} } {\ln u} = \dfrac 1 u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\ln u}
      | r = \map {\frac \d {\d u} } {\ln u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 u \frac {\d u} {\d x}
      | c = Derivative of Natural Logarithm Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Constant to Power of Function

:$\map {\dfrac \d {\d x} } {a^u} = a^u \ln a \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {a^u}
      | r = \map {\frac \d {\d u} } {a^u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = a^u \ln a \frac {\d u} {\d x}
      | c = Derivative of Power of Constant
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Exponential of Function

:$\map {\dfrac \d {\d x} } {e^u} = e^u \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {e^u}
      | r = \map {\frac \d {\d u} } {e^u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = e^u \frac {\d u} {\d x}
      | c = Derivative of Exponential Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Arcsine of Function

:$\map {\dfrac \d {\d x} } {\arcsin u} = \dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsin u}
      | r = \map {\frac \d {\d u} } {\arcsin u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}
      | c = Derivative of Arcsine Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Arccosine of Function

:$\map {\dfrac \d {\d x} } {\arccos u} = -\dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$

Proof:
{{:Graph of Arccosine Function}}
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccos u}
      | r = \map {\frac \d {\d u} } {\arccos u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = -\dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}
      | c = Derivative of Arccosine Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Arctangent of Function

:$\map {\dfrac \d {\d x} } {\arctan u} = \dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arctan u}
      | r = \map {\frac \d {\d u} } {\arctan u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {1 + u^2} \frac {\d u} {\d x}
      | c = Derivative of Arctangent Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Arccotangent of Function

:$\map {\dfrac \d {\d x} } {\arccot u} = -\dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccot u}
      | r = \map {\frac \d {\d u} } {\arccot u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = -\dfrac 1 {1 + u^2} \frac {\d u} {\d x}
      | c = Derivative of Arccotangent Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Arcsecant of Function

:$\map {\dfrac \d {\d x} } {\arcsec u} = \dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsec u}
      | r = \map {\frac \d {\d u} } {\arcsec u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\size u \sqrt {u^2 - 1} } \frac {\d u} {\d x}
      | c = Derivative of Arcsecant Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Arccosecant of Function

:$\map {\dfrac \d {\d x} } {\arccsc u} = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccsc u}
      | r = \map {\frac \d {\d u} } {\arccsc u} \frac {\d u} {\d x}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \frac {\d u} {\d x}
      | c = Derivative of Arccosecant Function
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Even Function is Odd

Let $f$ be a differentiable real function such that $f$ is even.
Then its derivative $f'$ is an odd function.

Proof:
{{begin-eqn}}
{{eqn | l = \map f x
      | r = \map f {-x}
      | c = {{Defof|Even Function}}
}}
{{eqn | ll= \leadsto
      | l = \frac \d {\d x} \map f x
      | r = \frac \d {\d x} \map f {-x}
      | c = differentiating both sides {{WRT|Differentiation}} $x$
}}
{{eqn | ll= \leadsto
      | l = \map {f'} x
      | r = \map {f'} {-x} \times \paren {-1}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = -\map {f'} {-x}
      | c = 
}}
{{end-eqn}}
Hence the result by definition of odd function.
{{qed}}
Category:Even Functions
Category:Differential Calculus
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Theorem: Derivative of Odd Function is Even

Let $f$ be a differentiable real function such that $f$ is odd.
Then its derivative $f'$ is an even function.

Proof:
{{begin-eqn}}
{{eqn | l = \map f x
      | r = -\map f {-x}
      | c = {{Defof|Odd Function}}
}}
{{eqn | ll= \leadsto
      | l = \frac \d {\d x} \map f x
      | r = -\frac \d {\d x} \map f {-x}
      | c = differentiating both sides {{WRT|Differentiation}} $x$
}}
{{eqn | ll= \leadsto
      | l = \map {f'} x
      | r = -\map {f'} {-x} \times \paren {-1}
      | c = Chain Rule for Derivatives
}}
{{eqn | r = \map {f'} {-x}
      | c = 
}}
{{end-eqn}}
Hence the result by definition of even function.
{{qed}}
Category:Odd Functions
Category:Differential Calculus
efatzudp5edp8236s4w6wm3n5kyic8k
Theorem: Form of Prime Sierpiński Number of the First Kind

Suppose $S_n = n^n + 1$ is a prime Sierpiński number of the first kind.
Then:
:$n = 2^{2^k}$
for some integer $k$.

Proof:
{{AimForCont}} $n$ has an odd divisor $d$.
By Sum of Two Odd Powers:
:$\paren {n^{n/d} + 1} \divides \paren {\paren {n^{n/d}}^d + 1^d} = S_n$
thus $S_n$ is composite, which is a contradiction.
Hence $n$ has no odd divisors.
That is, $n$ is a power of $2$.
Write $n = 2^m$.
{{AimForCont}} that $m$ has an odd divisor $f$.
By Sum of Two Odd Powers:
:$\paren {2^{n m/f} + 1} \divides \paren {\paren {2^{n m/f}}^f + 1^f} = 2^{m n} + 1 = S_n$
thus $S_n$ is composite, which is a contradiction.
Hence $m$ has no odd divisors.
That is, $m$ is a power of $2$.
Therefore we must have:
:$n = 2^{2^k}$
for some integer $k$.
{{qed}}
Theorem: Motion of Body Falling through Air

The motion of a body $B$ falling through air can be described using the following differential equation:
:$m \dfrac {\d^2 y} {\d t^2} = m g - k \dfrac {d y} {d t}$
where:
:$m$ denotes mass of $B$
:$y$ denotes the height of $B$ from an arbitrary reference
:$t$ denotes time elapsed from an arbitrary reference
:$g$ denotes the local gravitational constant acting on $B$
:$k$ denotes the coefficient of resistive force exerted on $B$ by the air (assumed to be proportional to the speed of $B$)

Proof:
From Newton's Second Law of Motion, the force on $B$ equals its mass multiplied by its acceleration.
Thus the force $F$ on $B$ is given by:
:$F = m \dfrac {\d^2 y} {\d t^2}$
where it is assumed that the acceleration is in a downward direction.
The force on $B$ due to gravity is $m g$.
The force on $B$ due to the air it is passing through is $k$ multiplied by the speed of $B$, in the opposite direction to its travel.
That is::
:$k \dfrac {d y} {d t}$
Hence the required differential equation:
:$m \dfrac {\d^2 y} {\d t^2} = m g - k \dfrac {d y} {d t}$
{{qed}}
Theorem: Rational Numbers are Everywhere Dense in Set of Real Numbers/Normed Vector Space

Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers.
Let $\Q$ be the set of rational numbers.
Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$

Proof:
{{WIP|Under brief review}}
We have that Between two Real Numbers exists Rational Number:
:$\forall a, b \in \R : a < b : \exists r \in \Q : a < r < b$
Let $a := x$ with $x \in \R$.
Let $\epsilon \in \R_{\mathop > 0} : r - a < \epsilon$.
Let $b := x + \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = x - \epsilon
      | o = <
      | r = x
}}
{{eqn | o = <
      | r = r
}}
{{eqn | o = <
      | r = x + \epsilon
}}
{{end-eqn}}
Since this holds for all $x$, we have that:
:$\forall x \in \R : \forall \epsilon \in \R_{\mathop > 0} : \exists r \in \Q : \size {x - r} < \epsilon$
By definition, $\Q$ is dense in $\R$.
Theorem: Number of Parameters of Autoregressive Model

Let $S$ be a stochastic process based on an equispaced time series.
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$
Let $M$ be an '''autoregessive model''' on $S$ of order $p$:
:$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$
Then $M$ has $p + 2$ parameters.
</onlyinclude>

Proof:
By definition of the parameters of $M$:
{{:Definition:Parameter of Autoregressive Model}}
Thus:
:there are $p$ parameters of the form $\phi_j$
:$1$ parameter $\mu$
:$1$ parameter $\sigma_a^2$.
That is: $p + 1 + 1 = p + 2$ parameters.
{{qed}}
Theorem: Autoregressive Model is Special Case of Linear Filter Model

Let $S$ be a stochastic process based on an equispaced time series.
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$
Let $M$ be an '''autoregessive model''' on $S$ of order $p$:
:$(1): \quad \tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$
Then $M$ is a special case of a linear filter model.
</onlyinclude>

Proof:
We can eliminate $\tilde z_{t - 1}$ from the {{RHS}} of $(1)$ by substituting:
:$\tilde z_{t - 1} = \phi_1 \tilde z_{t - 2} + \phi_2 \tilde z_{t - 3} + \dotsb + \phi_p \tilde z_{t - p - 1} + a_{t - 1}$
Similarly we can substitute for $\tilde z_{t - 2}$, and so on.
Eventually we get an infinite series in $a_{t - j}$.
Hence:
:$\map \phi B \tilde z_t = a_t$
is equivalent to:
:$\tilde z_t = \map \psi B a_t$
such that:
:$\map \psi B = \map {\phi^{-1} } B$
Hence the result by definition of linear filter model.
{{qed}}