Derivative of √x

The formula quietly predicts the square root's most famous behavior: as x → 0⁺ the slope 1/(2√x) blows up to infinity, which is why the graph leaves the origin vertically. And as x grows, the slope decays toward zero — the root keeps rising forever but ever more slowly. One derivative, both ends of the story.

This is also the standard demonstration that the power rule isn't just for whole numbers. On positive inputs, it works for any real exponent — halves, thirds, negatives, even irrational powers like x^π — which collapses square roots, cube roots, and reciprocals into a single rule instead of three separate formulas.

The real square-root function is defined for x ≥ 0, but its derivative 1/(2√x) is finite only for x > 0. At zero, the right-hand difference quotient is √h/h = 1/√h, which tends to positive infinity as h approaches zero. This gives a vertical tangent, not a finite derivative at zero.

For √g(x), the chain-rule formula is g′(x)/(2√g(x)) at points where g is differentiable and g(x) > 0. Points where g(x) = 0 require separate analysis. For example, √(x²) = |x| has derivative −1 for x < 0 and 1 for x > 0, but no derivative at zero.

For √(x² + 1), the inner expression is always positive, so the derivative x/√(x² + 1) exists for all real x. The power rule for a general real exponent is safely applied on positive inputs; other domains depend on the exponent.