AP Calculus AB and BC

Derivative of x^5: The Power Rule

The derivative of x to the fifth is 5x to the fourth. The power rule brings the exponent down in front and reduces it by one. The derivative is zero at the origin but never negative, so the origin is a flat spot rather than a maximum or minimum.

\frac{d}{dx}\left[x^{5}\right] = 5x^{4}

The power rule

\frac{d}{dx}x^{n} = nx^{n-1} \implies \frac{d}{dx}x^{5} = 5x^{4}

A zero derivative without an extremum

At $x = 0$ the derivative is $0$ , but $5x^{4}$ is positive on both sides, so there is no sign change and therefore no extremum. The function is increasing everywhere; it just pauses.

The second derivative $20x^{3}$ DOES change sign at the origin, so $x = 0$ is an inflection point. Every odd power above $1$ behaves this way.

Common mistakes

Assuming a zero derivative means a maximum or minimum. It requires a SIGN CHANGE.
Answering $5x^{5}$ or $x^{4}$ , forgetting to lower the exponent or to multiply by it.

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of x^5?

It is $5x^{4}$ .

Is x = 0 a maximum or minimum?

Neither. The derivative is zero there but does not change sign, so it is a flat inflection point.