AP Calculus AB and BC

Derivative of x^6: The Power Rule

Q: Is x = 0 a minimum?

Yes. 6x^{5} changes from negative to positive there, so the First Derivative Test gives a minimum.

Q: How does x^5 differ?

Its derivative 5x^{4} never changes sign, so the origin is a flat inflection point rather than an extremum.

The derivative of x to the sixth is 6x to the fifth. The power rule multiplies by the exponent and lowers it by one. Because the original exponent is even, the derivative has an odd exponent and so changes sign at the origin, which makes x equals 0 a minimum.

\frac{d}{dx}\left[x^{6}\right] = 6x^{5}

The power rule

\frac{d}{dx}x^{n} = nx^{n-1} \implies \frac{d}{dx}x^{6} = 6x^{5}

The derivative is negative for $x < 0$ and positive for $x > 0$ , so the function falls then rises and the origin is an absolute minimum.

Even and odd powers behave differently

An even power always gives an odd derivative, which changes sign at $0$ and produces an extremum. An odd power gives an even derivative, which does not change sign, so $x^{5}$ has a flat inflection point at the origin rather than a minimum.

The second derivative here is $30x^{4}$ , non-negative everywhere, so the curve is concave up on both sides and never inflects.

Common mistakes

Answering $6x^{6}$ or $x^{5}$ , forgetting to lower the exponent or to multiply by it.
Assuming every zero derivative is an extremum. It is here, but only because $6x^{5}$ actually changes sign.

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^6?

It is $6x^{5}$ .

Is x = 0 a minimum?

Yes. $6x^{5}$ changes from negative to positive there, so the First Derivative Test gives a minimum.

How does x^5 differ?

Its derivative $5x^{4}$ never changes sign, so the origin is a flat inflection point rather than an extremum.