AP Calculus AB and BC

Derivative of x^2: Answer, Proof, and Mistakes

Q: What is the derivative of x^2?

It is 2x. The power rule multiplies by the exponent 2 and lowers the exponent to 1.

Q: What is the second derivative of x^2?

It is 2. Differentiating 2x gives the constant 2, which is positive everywhere, so x^{2} is concave up everywhere.

Q: What is the derivative of x^2 at x = 3?

It is 6. Substitute into the derivative 2x AFTER differentiating, never before.

The derivative of x squared with respect to x is 2x. The power rule brings the exponent down in front and reduces it by one. Because the answer still contains x, the slope of the parabola is different at every point: it is 0 at the vertex and grows steeper as you move away.

\frac{d}{dx}\left[x^{2}\right] = 2x

How to differentiate x^2

The power rule handles this in one step: multiply by the exponent, then subtract one from it.

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

With $n = 2$ you get $2x^{1}$ , which is just $2x$ .

\frac{d}{dx}x^{2} = 2x

Prove it from the definition

Expand the difference quotient: $\frac{(x+h)^{2}-x^{2}}{h} = \frac{2xh + h^{2}}{h} = 2x + h$ . Letting $h \to 0$ leaves $2x$ . The $x^{2}$ terms cancel, which is the whole trick.

What 2x actually tells you

The derivative is a function, not a number, so it reports a different slope at every input. At $x = 0$ the slope is $0$ , which is the vertex of the parabola; at $x = 3$ it is $6$ .

The derivative is negative for $x < 0$ and positive for $x > 0$ , which says the parabola falls then rises, and the sign change at $0$ is exactly the First Derivative Test finding the minimum.

The second derivative is the constant $2$ . It is positive everywhere, so the graph is concave up everywhere and has no inflection point.

Where the derivative of x^2 shows up on the AP exam

Basic derivative rules are Topic 2.5 and the definition is Topic 2.1, both on AB and BC. This is also the function most often used to test the difference quotient itself, because the algebra is short enough to do under time pressure.

Watch for the chain rule variants, which are different problems: $\frac{d}{dx}\left(2x\right)^{2} = 8x$ , and $\frac{d}{dx}x^{2}\sin x = 2x\sin x + x^{2}\cos x$ by the product rule.

Common mistakes with the derivative of x^2

Answering $2$ . That is the derivative of $2x$ , and it is also the SECOND derivative of $x^{2}$ , so it is a plausible wrong answer.
Answering $x$ . Dividing the exponent instead of multiplying by it is the antiderivative pattern, not the derivative one.
Treating $x^{2}$ like $2^{x}$ . The variable is in the base here, so the power rule applies; with the variable in the exponent the derivative is $2^{x}\ln 2$ .
Substituting a number before differentiating. Plugging in $x = 3$ first makes the function the constant $9$ , whose derivative is $0$ .

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^2?

It is $2x$ . The power rule multiplies by the exponent $2$ and lowers the exponent to $1$ .

Why is the derivative of x^2 not a constant?

Because the slope of a parabola genuinely changes from point to point. Only a straight line has a constant derivative.

What is the second derivative of x^2?

It is $2$ . Differentiating $2x$ gives the constant $2$ , which is positive everywhere, so $x^{2}$ is concave up everywhere.

What is the derivative of x^2 at x = 3?

It is $6$ . Substitute into the derivative $2x$ AFTER differentiating, never before.