Derivative of x^3: Answer, Proof, and Mistakes

How to differentiate x^3

The power rule differentiates any power of $x$ in one step: multiply by the exponent, then subtract one from it.

With $n = 3$ the exponent $3$ moves to the front and drops to $2$.

What 3x^2 tells you about the graph

The derivative is a function, so it reports a different slope at every input. At $x = 2$ the slope is $12$, and at $x = -2$ it is also $12$, since squaring removes the sign.

Because $3x^{2} \ge 0$ for all $x$, the cubic is increasing everywhere except at $x = 0$, where the tangent is horizontal. That single flat spot is a stationary inflection point, not an extremum, because the slope does not change sign around it.

The second derivative is $6x$, negative for $x < 0$ and positive for $x > 0$. The graph is concave down then concave up, and the concavity changes exactly at the origin.

Where the derivative of x^3 shows up on the AP exam

The power rule is Topic 2.5 (Applying the Power Rule), on both AB and BC. Unit 2 carries 10 to 15 percent of the AB exam weighting and 5 to 10 percent on BC, and $x^{3}$ is one of the first powers you practice it on.

It rarely appears bare. More often $x^{3}$ sits inside a chain rule such as $\frac{d}{dx}(x^{3}+1)^{5}$, or inside a product like $x^{3}e^{x}$, so recognizing $3x^{2}$ instantly frees you to focus on the harder structure.

Common mistakes with the derivative of x^3

• Answering $x^{2}$. Subtracting one from the exponent is correct, but you also multiply by the old exponent, so the coefficient is $3$, not $1$.
• Answering $3x^{3}$. Multiplying by the exponent without lowering it leaves the power unchanged.
• Answering $6x$. That is the second derivative of $x^{3}$, a common distractor.
• Treating $x^{3}$ like $3^{x}$. The variable is in the base here, so the power rule applies; with the variable in the exponent the derivative is $3^{x}\ln 3$.