What is the Second Derivative?
When you differentiate a function once, you get the first derivative , which describes the rate of change of the function. Differentiating again gives you the second derivative , a measure of how the rate of change itself is changing.
Second Derivative: The second derivative is obtained by differentiating a function twice. It describes how the gradient of a function is changing at any given point.
There are two standard notations you'll encounter in IB exams:
- Leibniz notation:
- Lagrange notation:
Finding the second derivative
Let .
Step 1: Differentiate once:
Step 2: Differentiate again:
So at , the second derivative is , meaning the gradient of is increasing at that point.
In Leibniz notation, does not mean . The superscript 2 indicates the order of differentiation, not a power.
Concavity: What the Second Derivative Tells Us About Shape
The most important graphical interpretation of the second derivative is concavity , describing whether a curve bends upward or downward.
Concavity: Concavity describes the way a curve bends. A function is concave up when its gradient is increasing, and concave down when its gradient is decreasing.
The sign of determines concavity at any point:
| Concavity | Shape | |
|---|---|---|
| Concave up | (like a cup) | |
| Concave down | (like an arch) |
Think of concavity in terms of a bowl of water: if the curve is concave up (), the bowl holds water. If it's concave down (), the water spills off. The second derivative tells you which way the bowl faces.
Concavity is a local property , a function can be concave up in one region and concave down in another. You must always specify the interval or point you are discussing.