Derivative of x^3

Answer

3 \cdot x^{2}

Derivative of x^3

Rule Used: Power Rule
Steps: Apply the power rule: $\frac{d}{dx}$ ( $x^3$ ) = $3\cdot x^{3-1}$ . The exponent counts how many factors of x are multiplied, so one factor is lost and the count moves in front.
Simplify the exponent: $3\cdot x^2$
Final answer: $3 \cdot x^2$

d/dx x^3 = 3*x^2. The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

Instant step-by-step solution for x^3

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

Derivative

x^3 → 3*x^2engine: 3*x^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

x^3 needs the binomial theorem, not the square case copied forward. (x+h)^3 = x^3 + 3x^2 h + 3x h^2 + h^3, so after subtracting x^3 and dividing by h the terms with h vanish and 3x^2 remains. The 3 is the number of ways to pick which factor of x supplied the h.

Apply the power rule to x^3. The form to remember, already simplified, is 3*x^2.

The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

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

Worked derivative

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

Common mistakes

Answering 2*x^2

2x^2 looks plausible if you remember x^2 but forget that the front factor is the old exponent, which is 3 here, not 2.

Forgetting to lower the exponent of x^3

The front coefficient of x^3 is only half the work. The exponent itself has to decrease by 1. Leaving the original power in place describes a different curve.

Power rule — The rule page lists this example and the others in the same family.
Derivative of x^2 — Same family of differentiation, different function.
Derivative of x^4 — Same family of differentiation, different function.
Derivative of 3x^2 — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of x^3?

3*x^2. x^3 needs the binomial theorem, not the square case copied forward. (x+h)^3 = x^3 + 3x^2 h + 3x h^2 + h^3, so after subtracting x^3 and dividing by h the terms with h vanish and 3x^2 remains. The 3 is the number of ways to pick which factor of x supplied the h.