Derivative of x³
One notch up from x², the cube is where students learn that the power rule scales — and where the derivative stops being a straight line.
Step-by-step solution
f(x) = x3: variable base, constant exponent n = 3.
Bring the 3 down and knock the exponent from 3 to 2: 3·x3−1.
f′(x) = 3x2.
Why it works
3x² is never negative, and that single observation tells you the entire story of the cubic's shape: x³ is increasing everywhere. It does flatten to slope zero at exactly one point (the origin), but it never actually turns around — which makes x = 0 the textbook example of a critical point that is not a max or min. It's an inflection point, where the curve pauses and continues climbing.
There's also a neat volume interpretation. If x is the side of a cube, x³ is its volume, and 3x² is the surface area of three of its faces — exactly the material you'd add by growing the cube slightly along its three dimensions. The power rule, seen this way, is bookkeeping for how an n-dimensional box grows.
Stationary point and tangent lines
X³ and its derivative 3x² exist for every real x. At zero the derivative vanishes, but it is positive on both sides, so zero is not a local maximum or minimum. The second derivative 6x changes sign there, making the origin an inflection point with tangent line y = 0.
At x = −1 the value is −1 and the slope is 3. Its tangent line is y = −1 + 3(x + 1). The same slope occurs at x = 1, although the graph point and tangent line differ.
Common mistakes
- Writing 3x³ (forgetting to reduce the exponent) or x² (forgetting the coefficient).
- Concluding x = 0 is a minimum because the derivative is zero there — the first-derivative test shows the slope is positive on both sides.
- Sloppy negatives on (−x)³: its derivative is −3x², since (−x)³ = −x³.
Practice problems
Differentiate 2x³ + x
Answer: 6x² + 1.
Find the second derivative of x³
Answer: 6x — differentiate 3x² again.
Where does x³ have slope 27?
Answer: 3x² = 27 → x = ±3.