sqrtX

Sqrtx, written sqrt(x) or x^(1/2), denotes the principal square root function. It returns the nonnegative number y such that y^2 = x. In the real numbers, sqrt(x) is defined only for x ≥ 0; for x < 0 the real square root does not exist, though complex numbers provide square roots as well.

Notation and basic properties: For x ≥ 0, sqrt(x) ≥ 0 and sqrt(a^2) = |a|. The product rule sqrt(ab) = sqrt(a) sqrt(b) holds when a ≥ 0 and b ≥ 0. The function is increasing on [0, ∞) and concave on (0, ∞).

Calculus: The derivative is d/dx sqrt(x) = 1/(2 sqrt(x)) for x > 0, and the integral ∫ sqrt(x) dx = (2/3) x^(3/2) + C.

Complex extension: In the complex plane, every nonzero z has two square roots. A principal value is often defined by sqrt(z) = exp((1/2) Log z) with the principal branch cut, typically along the negative real axis.

Examples and applications: sqrt(4) = 2, sqrt(2) ≈ 1.4142, sqrt(0) = 0. The square root is used to solve equations, compute areas, and model magnitude in physics and engineering. Numerical methods such as the Newton-Raphson iteration are used to approximate sqrt(x) for general x.

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