Derivative of Root x (original) (raw)

Last Updated : 17 Apr, 2026

The derivative of \sqrt{x}, with respect to x is

\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}

There are two methods to find the derivative of a root x:

Derivative of Root x Using First Principle

First principle of differentiation state that derivative of a function f(x) is defined as,

**f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{(x+h) - x}

**f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}

Putting f(x) = √x, to find derivative of root x, we get,

f'(x) = limh→0 [√(x + h) – √(x)]/ h

Multiplying numerator and denominator by √(x + h) + √(x), we get,

⇒ limh→0 [√(x + h) – √(x)]×[√(x + h) + √(x)]/[h×(√(x + h) + √(x))]

⇒ limh→0 [|x+h-x|] / [h×(√(x + h) + √(x))]

⇒ limh→0 [h] / [h×(√(x + h) + √(x))]

⇒ limh→0 1/ [(√(x + h) + √(x))]

⇒ 1/ [(√(x + 0) + √(x))]

⇒ 1/2√x

Derivative of Root x Using Power Rule

Root x is an algebraic function which can be represented as x1/2. Applying the power rule to find derivative of x1/2, we get,

(x1/2)' = 1/2(x)1/2-1

⇒ 1/2(x)-1/2

**⇒ 1/2x1/2 or 1/2√x

Thus, we derived the derivative of root x using both, First Principle approach and Power Rule.

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