Derivative of x^4

Answer

4 \cdot x^{3}

Derivative of x^4

Rule Used: Power Rule
Steps: Apply the power rule: $\frac{d}{dx}$ ( $x^4$ ) = $4\cdot x^{4-1}$ . The exponent counts how many factors of x are multiplied, so one factor is lost and the count moves in front.
Simplify the exponent: $4\cdot x^3$
Final answer: $4 \cdot x^3$

d/dx x^4 = 4*x^3. 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^4

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Derivative of x^4 = 4*x^3 Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-x-to-the-4

Derivative

x^4 → 4*x^3engine: 4*x^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

By the time the power is 4, the pattern is the lesson rather than another binomial expansion. Each of the four factors of x can be the one that produces the small change, so the derivative is 4x^3. Differentiating again would give 12x^2, which is the second-derivative check that the first factor really was 4.

Apply the power rule to x^4. The form to remember, already simplified, is 4*x^3.

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

At x = 1, substitute into the simplified form 4*x^3. That number is the slope of x^4 there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = x^4.
Apply the power rule to x^4. The form to remember, already simplified, is 4*x^3.
At x = 1, substitute into the simplified form 4*x^3. That number is the slope of x^4 there; a secant over a small step in x lands next to it.

Common mistakes

Answering 3*x^3

3x^3 would be right for x^3, and it is the usual off-by-one when the exponent and the front coefficient get swapped in your head.

Forgetting to lower the exponent of x^4

The front coefficient of x^4 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^3 — 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^4?

4*x^3. By the time the power is 4, the pattern is the lesson rather than another binomial expansion. Each of the four factors of x can be the one that produces the small change, so the derivative is 4x^3. Differentiating again would give 12x^2, which is the second-derivative check that the first factor really was 4.