Derivative of sqrt(x)

Answer

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

Derivative of sqrt(x)

Rule Used: Power Rule
Steps: Rewrite sqrt as a power: $\sqrt{u}$ = $u^{1/2}$ , then apply the power rule. The new exponent is 1/2 − 1 = −1/2, and the 1/2 stays in front.
Final answer: (1/2)* $x^{-1/2}$

d/dx sqrt(x) = (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 sqrt(x)

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

Derivative

sqrt(x) → (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

Square root is the power x^(1/2). The power rule multiplies by 1/2 and subtracts 1 from the exponent, leaving (1/2) x^(−1/2), which is 1/(2 sqrt(x)). The 2 in the denominator is the old exponent turned upside down; it is not an extra chain-rule factor.

Apply the power rule to sqrt(x). 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 = 4, substitute into the simplified form (1/2)*x^(-1/2). That number is the slope of sqrt(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = sqrt(x).
Apply the power rule to sqrt(x). The form to remember, already simplified, is (1/2)*x^(-1/2).
At x = 4, substitute into the simplified form (1/2)*x^(-1/2). That number is the slope of sqrt(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering 1/(2x) instead of 1/(2√x)

1/(2x) puts the 2 next to x instead of next to the square root. At x = 4 that wrong answer is 1/8, while the true slope is 1/4.

Ignoring the domain of sqrt(x)

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 x^(1/2) — Same family of differentiation, different function.
Derivative of cbrt(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of sqrt(x)?

(1/2)*x^(-1/2). Square root is the power x^(1/2). The power rule multiplies by 1/2 and subtracts 1 from the exponent, leaving (1/2) x^(−1/2), which is 1/(2 sqrt(x)). The 2 in the denominator is the old exponent turned upside down; it is not an extra chain-rule factor.