Derivative of x^(1/2)

Answer

\frac{1}{2 \cdot x^{\frac{1}{2}}}

Derivative of x^(1/2)

Rule Used: Power Rule
Steps: Identify the expression as a power function and apply the power rule. A rational exponent is allowed: $\frac{d}{dx} x$ ^(p/q) = (p/q)· $x^{p/q - 1}$ .
Final answer: (1/2)* $x^{-1/2}$

d/dx x^(1/2) = (1/2)*x^(-1/2). The expression needs x > 0. At 0 and on the negatives it is undefined as a real function, so the derivative is too.

Instant step-by-step solution for x^(1/2)

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Derivative of x^(1/2) = (1/2)*x^(-1/2) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-root-x

Derivative

x^(1/2) → (1/2)*x^(-1/2)engine: (1/2)*x^(-1/2)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

This page writes the square root as the rational exponent 1/2 instead of the radical sign. The algebra is the same function as sqrt(x), and the derivative matches, but the steps start from n = 1/2: multiply by 1/2, then the new exponent is 1/2 − 1 = −1/2. Students who searched the exponent form need to see that subtraction, not a rewritten radical, as the first move.

Apply the power rule to x^(1/2). The form to remember, already simplified, is (1/2)*x^(-1/2).

The expression needs x > 0. At 0 and on the negatives it is undefined as a real function, so the derivative is too.

At x = 9, substitute into the simplified form (1/2)*x^(-1/2). That number is the slope of x^(1/2) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = x^(1/2).
Apply the power rule to x^(1/2). The form to remember, already simplified, is (1/2)*x^(-1/2).
At x = 9, substitute into the simplified form (1/2)*x^(-1/2). That number is the slope of x^(1/2) there; a secant over a small step in x lands next to it.

Common mistakes

Answering (1/2)*x^(1/2)

Multiplying by 1/2 without changing the exponent leaves (1/2) sqrt(x), which still grows, while the true derivative shrinks as x gets large.

Ignoring the domain of x^(1/2)

A fractional power can blow up at 0 or refuse negative inputs. The algebraic formula is not a license to evaluate it where the original root is not real.

Power rule — The rule page lists this example and the others in the same family.
Derivative of sqrt(x) — Same family of differentiation, different function.
Derivative of cbrt(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of x^(1/2)?

(1/2)*x^(-1/2). This page writes the square root as the rational exponent 1/2 instead of the radical sign. The algebra is the same function as sqrt(x), and the derivative matches, but the steps start from n = 1/2: multiply by 1/2, then the new exponent is 1/2 − 1 = −1/2. Students who searched the exponent form need to see that subtraction, not a rewritten radical, as the first move.