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t ) = f ( x , b ( x ) ) ⋅ d d x b ( x ) − f ( x , a ( x ) ) ⋅ d d x a ( x ) + ∫ a ( x ) b ( x ) ∂ ∂ x f ( x , t ) d t {\displaystyle {\begin{aligned}&{\frac...

f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + h 1 ( x ) ( x − a ) , lim x → a h 1 ( x ) = 0. {\displaystyle f(x)=f(a)+f'(a)(x-a)+h_{1}(x)(x-a),\quad \lim...

1 ] , S y = 1 2 det [ A x | A | 2 1 B x | B | 2 1 C x | C | 2 1 ] , a = det [ A x A y 1 B x B y 1 C x C y 1 ] , b = det [ A x A y | A | 2 B x B y | B...

two-dimensional acceleration vector is then defined as a =< a x , a y > {\displaystyle {\textbf {a}}=<a_{x},a_{y}>} . The magnitude of this vector is found by the...

lim x → a f ( g ( x ) ) − f ( g ( a ) ) x − a . {\displaystyle (f\circ g)'(a)=\lim _{x\to a}{\frac {f(g(x))-f(g(a))}{x-a}}.} Assume for the moment that g...

{\bmod {\,}}n} . Calculate the curve point Q A = ( x A , y A ) = u 1 × G + u 2 × R {\displaystyle Q_{A}=(x_{A},y_{A})=u_{1}\times G+u_{2}\times R} . The...

arbitrary nonzero point a is: ln ⁡ a + 1 a ( x − a ) − 1 a 2 ( x − a ) 2 2 + ⋯ . {\displaystyle \ln a+{\frac {1}{a}}(x-a)-{\frac {1}{a^{2}}}{\frac...

compound assignment operators (x += a, x -= a, x *= a, x /= a, x %= a, x <<= a, x >>= a, x &= a, x ^= a, x |= a) as in C and descendants. The preprocessor...

combining Raoult's law with Dalton's law of partial pressures to give p = p A ⋆ x A + p B ⋆ x B + ⋯ . {\displaystyle p=p_{\text{A}}^{\star }x_{\text{A}}+p_{\text{B}}^{\star...

  v = e a x a   {\displaystyle {\frac {du}{dx}}\ =nx^{n-1}{\text{ , }}\ v={\frac {e^{ax}}{a}}\ } I n = x n e a x a   − ∫ n x n − 1   e a x a   d x   {\displaystyle...

projection matrix: P a = a a T = [ a x a y a z ] [ a x a y a z ] = [ a x 2 a x a y a x a z a x a y a y 2 a y a z a x a z a y a z a z 2 ] {\displaystyle...

′ ( x ) − f ( x , a ( x ) ) a ′ ( x ) + ∫ a ( x ) b ( x ) ∂ ∂ x f ( x , t ) d t . {\displaystyle F'(x)=f(x,b(x))\,b'(x)-f(x,a(x))\,a'(x)+\int _{a(x)}^{b(x)}{\frac...

2ax-{\frac {x}{4a^{2}}}\cos 2ax+C} ∫ x sin ⁡ a x d x = sin ⁡ a x a 2 − x cos ⁡ a x a + C {\displaystyle \int x\sin ax\,dx={\frac {\sin ax}{a^{2}}}-{\frac...

{\displaystyle F(x_{1}+x_{2})=F(x_{1})+F(x_{2})} and homogeneity F ( a x ) = a F ( x ) {\displaystyle F(ax)=aF(x)} for scalar a. This principle has many...

invariant in the sense that for any real number a {\displaystyle a} , x + a ¯ = x ¯ + a {\displaystyle {\overline {x+a}}={\bar {x}}+a} . If it is required...

statements hold when A is a full rank square matrix: A^-1 *(A * x)==A^-1 * (b) (A^-1 * A)* x ==A^-1 * b       (matrix-multiplication associativity) x = A^-1...

{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}} d d x a x = a x ln ⁡ ( a ) {\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a)} , for a > 0 {\displaystyle a>0} d d...

{\displaystyle x\in V} and a ∈ F {\displaystyle a\in F} (this results from ⟨ a x , a x ⟩ = a a ¯ ⟨ x , x ⟩ {\displaystyle \langle ax,ax\rangle =a{\overline {a}}\langle...

cos ⁡ β 0 − sin ⁡ β 0 1 0 sin ⁡ β 0 cos ⁡ β ] [ a x a y a z ] = 1 6 [ 3 0 − 3 1 2 1 2 − 2 2 ] [ a x a y a z ] {\displaystyle {\begin{bmatrix}\mathbf {c}...

ci occupies a range from 1 to ( n . s t a t e s − 1 ) / ( n . t a x a − ⌈ n . t a x a / n . s t a t e s ⌉ ) {\displaystyle (n.states-1)/(n.taxa-\lceil...