formulas in different areas of mathematics:
Algebra:
Quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a
Binomial theorem: (a + b)^n = ∑(k=0 to n) (n choose k) a^(n-k) b^k
Factorization formulas: a^2 - b^2 = (a+b)(a-b), a^3 + b^3 = (a+b)(a^2-ab+b^2)
Geometry:
Pythagorean theorem: a^2 + b^2 = c^2 (for a right triangle where c is the hypotenuse)
Area of a circle: A = πr^2
Volume of a sphere: V = (4/3)πr^3
Calculus:
Derivative formulas: (d/dx)(f+g) = df/dx + dg/dx, (d/dx)(fg) = f(dg/dx) + g(df/dx), (d/dx)(f/g) = (g(df/dx) - f(dg/dx)) / g^2
Integration formulas: ∫k dx = kx + C, ∫x^n dx = (x^(n+1))/(n+1) + C, ∫sin(x) dx = -cos(x) + C
Probability and Statistics:
Mean formula: μ = (Σx)/n
Variance formula: σ^2 = (Σ(x-μ)^2)/n
Normal distribution formula: f(x) = (1/(σ√(2π)))e^(-((x-μ)^2)/(2σ^2))
These are just a few examples of formulas in mathematics, and there are many more.
