Derivative of x^(1/3)

Answer

\frac{1}{3 \cdot x^{\frac{2}{3}}}

Derivative of x^(1/3)

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: (1/3)* $x^{-2/3}$

d/dx cbrt(x) = (1/3)*x^(-2/3). The derivative exists for every real x except 0, where a denominator vanishes or a negative power blows up.

Instant step-by-step solution for x^(1/3)

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Derivative of x^(1/3) = (1/3)*x^(-2/3) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-cbrt-x

Derivative

cbrt(x) → (1/3)*x^(-2/3)engine: (1/3)*x^(-2/3)

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

The cube root is x^(1/3), and unlike the square root it is defined for negative x. The power rule multiplies by 1/3 and the new exponent is 1/3 − 1 = −2/3, so the derivative is 1/(3 x^(2/3)). The exponent on the denominator is 2/3, not 1/3, because you subtracted 1 rather than copying the original power.

Apply the power rule to cbrt(x). The form to remember, already simplified, is (1/3)*x^(-2/3).

The derivative exists for every real x except 0, where a denominator vanishes or a negative power blows up.

At x = 8, substitute into the simplified form (1/3)*x^(-2/3). That number is the slope of cbrt(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = cbrt(x).
Apply the power rule to cbrt(x). The form to remember, already simplified, is (1/3)*x^(-2/3).
At x = 8, substitute into the simplified form (1/3)*x^(-2/3). That number is the slope of cbrt(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering (1/3)*x^(1/3)

Keeping the exponent at 1/3 after multiplying by 1/3 skips the subtraction. The derivative must blow up at zero; (1/3) cbrt(x) does not.

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

Frequently asked questions

What is the derivative of cbrt(x)?

(1/3)*x^(-2/3). The cube root is x^(1/3), and unlike the square root it is defined for negative x. The power rule multiplies by 1/3 and the new exponent is 1/3 − 1 = −2/3, so the derivative is 1/(3 x^(2/3)). The exponent on the denominator is 2/3, not 1/3, because you subtracted 1 rather than copying the original power.