Derivative of x^4: Answer, Proof, and Mistakes

How to differentiate x^4 with the power rule

The power rule differentiates any power $x^{n}$ by bringing the exponent down as a factor and reducing the exponent by one.

Here $n = 4$, so the exponent $4$ becomes the leading coefficient and the new exponent is $4 - 1 = 3$.

What the derivative tells you about x^4

Because $4x^{3}$ is an odd power scaled by a positive constant, the slope is negative for $x < 0$, zero at $x = 0$, and positive for $x > 0$. So $x^{4}$ falls then rises, with its only critical point at the origin.

The point $x = 0$ is a minimum, but a flat one: the second derivative $12x^{2}$ is also zero there, so the second derivative test is inconclusive and you read the behavior from the sign change of $4x^{3}$ instead.

The reverse operation raises the exponent by one and divides: $\int x^{4}\,dx = \frac{x^{5}}{5} + C$.

Where the derivative of x^4 shows up on the AP exam

The power rule is Topic 2.5, on both AB and BC, and a single power like $x^{4}$ is the plainest test of it. It also appears inside longer polynomials, where you differentiate term by term.

Common mistakes with the derivative of x^4

• Answering $4x^{4}$, attaching the coefficient but forgetting to lower the exponent to $3$.
• Answering $x^{3}$, lowering the exponent but forgetting to multiply by the exponent $4$.
• Answering $12x^{2}$, which is the second derivative, not the first.