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481.
Radius of a circle
If the circle equation is: $Ax^2+Ay^2+DX+Ey+F=0$ than the radius is equal: $R = \sqrt{\dfrac{D^2+E^2-4AF}{2|A|}}$
Analytic geometry
Circle
482.
$\lim_{x\to 0} \dfrac{\sin ax}{bx}=\dfrac{a}{b}$
Limit
Trigonometry
483.
$\lim_{x\to\infty} x \sin \left( \dfrac{1}{x} \right) = 1$
Limit
Trigonometry
484.
$\lim{x \to \infty}\left(1+\dfrac{k}{x}\right)^{mx}=e^{km}$
E
Limit
485.
Linearity of differentiation
$\left(af+bg\right)'=af'+bg'$
Differentiation
Differentiation rule
486.
$\dfrac{d}{dx}\, \left(ax^2+bx+c\right) = 2ax+b$
Differentiation
487.
$\dfrac{d}{dx}\, \left( x^n \right) = nx^{n-1}$
Differentiation
488.
$\left(f^n\right)'=n \cdot f^{n-1} \cdot f'$
Differentiation
489.
$\left(\sqrt{f}\right)' = \dfrac{f'}{2\sqrt{f}}$
Differentiation
490.
$\left( \ln f \right)' = \dfrac{f'}{f}$
Differentiation
491.
$\left( \log_a x \right)'=\dfrac{f'}{f \ln a}$
Differentiation
492.
n-th derivative of $e^x$
$\left(e^x\right)^{(n)}=e^x$
Differentiation
493.
$\int \sqrt{ax+b}dx=\dfrac{2}{3a}\left(ax+b\right)^{3/2}+C$
Indefinite Integral
Irrational Function
494.
$\int x\sqrt{ax+b}dx=\dfrac{2(3ax-2b)}{15a^2}(ax+b)^{3/2}+C$
Indefinite Integral
Irrational Function
495.
$\int \dfrac{dx}{(x+c)\sqrt{ax+b}}=\dfrac{1}{\sqrt{b-ac}} \ln\left|\dfrac{\sqrt{ax+b}-\sqrt{b-ac}}{\sqrt{ax+b}+\sqrt{b-ac}}\right|+C, b-ac>0$
Indefinite Integral
Irrational Function
496.
$\int \dfrac{dx}{(x+c)\sqrt{ax+b}}=\dfrac{1}{\sqrt{ac-b}}\arctan\sqrt{\dfrac{ax+b}{ac-b}}+C, b-ac<0$
Indefinite Integral
Irrational Function
497.
$\int x^2\sqrt{a+bx}dx=\dfrac{2\left(8a^2-12abx+15b^2x^2\right)}{105b^3}\sqrt{(a+bx)^3}+C$
Indefinite Integral
Irrational Function
498.
$\int \dfrac{x^2}{\sqrt{a+bx}} = \dfrac{2\left(8a^2-4abx+3b^2x^2\right)}{15b^3}\sqrt{a+bx}+C$
Indefinite Integral
Irrational Function
499.
Difference of arithmetic series
$d=a_n-a_{n-1}$
Arithmetic Series
500.
Sum of infinite geometric series
$S=\dfrac{a_1}{1-q}, \text{for} |q| < 1$
Geometric series

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