What Does the Second Derivative Represent?

The second derivative represents the rate at which a function's gradient is changing. It is written as (f''(x)) or $\frac{d^2y}{dx^2}$, and it describes concavity, helps classify stationary points, and can represent acceleration in a motion context.

How to Interpret the Second Derivative

Starting with $y=f(x)$, the first derivative (f'(x)) gives the gradient or instantaneous rate of change. Differentiating again gives (f''(x)), which shows whether that gradient is increasing or decreasing.

In context, the meaning depends on the variables. If $s(t)$ is displacement, then (s'(t)) is velocity and (s''(t)) is acceleration. A positive second derivative means velocity is increasing, not necessarily that the object is moving forward. Similarly, concave up does not mean the function itself is increasing; the graph may still decrease while its gradient becomes less negative. This distinction is frequently tested.

At a stationary point, where $f'(x)=0$, the sign of the second derivative can classify the point. If $f''(x)>0$, it is a local minimum; if , it is a local maximum.

For example, let

$f(x)=x^3-3x.$

Then

$f'(x)=3x^2-3, \qquad f''(x)=6x.$

The stationary points occur when $3x^2-3=0$, giving $x=-1$ and $x=1$. Since , the point at is a local maximum. Since , the point at is a local minimum.

A common misconception is that $f''(x)=0$ automatically identifies a point of inflexion. It does not: the concavity must change sign across that point. Here, $f''(x)=6x$ changes from negative to positive at , so there is an inflexion point there.

IB Exam Technique

This is HL-only content from AHL 5.10: Second derivatives, testing for max and min. In Paper 1 or Paper 2, show $f'(x)=0$, calculate (f''(x)), substitute the stationary-point value, and state the conclusion clearly; all Math AI papers are GDC-active.