function dist(t) {
if (t <= 2) return 5 * t * t;
if (t <= 3) return 20 * t - 20;
if (t <= 7) return -5 * t * t + 50 * t - 65;
return -20 * t + 180;
}
function vel(t) {
if (t <= 2) return 10 * t;
if (t <= 3) return 20;
if (t <= 7) return -10 * t + 50;
return -20;
}
function acc(t) {
if (t <= 2) return 10;
if (t <= 3) return 0;
if (t <= 7) return -10;
return 0;
}Second Derivative
calculus
In the previous page, you saw how the derivative of a derivative appeared naturally in Newton’s method for optimization. This “derivative of a…
In the previous page, you saw how the derivative of a derivative appeared naturally in Newton’s method for optimization. This “derivative of a derivative” has a name: the second derivative. It carries rich information about a function’s shape and is essential for determining whether a critical point is a maximum or a minimum.
Notation
In Leibniz notation, the second derivative is written as:
\[\frac{d^2 f}{dx^2} \quad \text{or equivalently} \quad \frac{d}{dx}\left(\frac{df}{dx}\right)\]
In Lagrange notation, it is simply:
\[f''(x)\]
If \(f'(x)\) is the first derivative (rate of change of \(f\)), then \(f''(x)\) is the rate of change of \(f'\), the rate of change of the rate of change.
Real-World Intuition: Distance, Velocity, Acceleration
To build intuition, recall that the first derivative of distance with respect to time is velocity:
\[v = \frac{dx}{dt}\]
The second derivative of distance is the rate of change of velocity, which is acceleration:
\[a = \frac{dv}{dt} = \frac{d^2 x}{dt^2}\]
Acceleration tells us how the velocity is changing:
- Positive acceleration: velocity is increasing (speeding up).
- Negative acceleration: velocity is decreasing (slowing down).
- Zero acceleration: constant velocity (no change in speed).
Driving Example
Imagine you are driving your car. The horizontal axis is time (in seconds) and the vertical axis is distance (in meters). You can drag the point along the time axis below to see the connection between the function and its derivatives, first and second.
Move your mouse over the chart to explore. Notice how:
- The tangent line (teal) on the distance plot shows the current slope, which equals the velocity.
- When the velocity is increasing, the acceleration is positive. When velocity is decreasing, acceleration is negative. When velocity is constant, acceleration is zero.
Now let us look at each phase with the actual formulas.
Phase 1: Speeding Up (0 to 2 seconds)
You start from rest and accelerate. The distance function is:
\[d(t) = 5t^2\]
The velocity (first derivative) is \(v(t) = 10t\), which increases linearly. The acceleration (second derivative) is \(a(t) = 10\), a positive constant. You are speeding up at a constant rate.
Phase 2: Constant Speed (2 to 3 seconds)
You reach cruising speed and maintain it. The distance function is linear:
\[d(t) = 20t - 20\]
The velocity is \(v(t) = 20\) m/s (constant). The acceleration is \(a(t) = 0\). No change in speed.
Phase 3: Slowing Down and Turning Back (3 to 7 seconds)
You realize you forgot something and hit the brakes. The distance function is:
\[d(t) = -5t^2 + 50t - 65\]
The velocity is \(v(t) = -10t + 50\), which decreases linearly, crosses zero at \(t = 5\) (you stop), then goes negative (you reverse). The acceleration is \(a(t) = -10\), a negative constant. You are decelerating.
Notice: at the moment you stop (maximum distance, \(t = 5\)), \(v = 0\) and \(a < 0\). The negative second derivative at a zero of the first derivative confirms this is a maximum.
Phase 4: Constant Speed in Reverse (7 to 9 seconds)
You drive back at a steady pace:
\[d(t) = -20t + 180\]
The velocity is \(v(t) = -20\) m/s (constant, negative means going back). The acceleration is \(a(t) = 0\).
Summary Table
| Phase | Distance | Velocity | Acceleration |
|---|---|---|---|
| Speeding up | Concave up (curving upward) | Increasing | Positive |
| Constant speed | Linear (straight line) | Constant | Zero |
| Slowing/reversing | Concave down (curving downward) | Decreasing | Negative |
| Constant reverse | Linear (straight line) | Constant (negative) | Zero |
Concavity
The second derivative measures curvature, how much a function deviates from being a straight line.
- Positive second derivative (\(f'' > 0\)): the function is concave up (curves like a bowl, or a happy face 😊). The slope is increasing.
- Negative second derivative (\(f'' < 0\)): the function is concave down (curves like a dome, or a sad face 😢). The slope is decreasing.
- Zero second derivative (\(f'' = 0\)): inconclusive. The function might be linear at that point, or it could be an inflection point.
Second Derivative Test for Optimization
This is the key application. At a critical point (where \(f'(x) = 0\)), the second derivative tells you what kind of extremum you have:
| \(f'(x) = 0\) and… | Conclusion |
|---|---|
| \(f''(x) > 0\) | Local minimum (concave up, bowl shape) |
| \(f''(x) < 0\) | Local maximum (concave down, dome shape) |
| \(f''(x) = 0\) | Inconclusive (could be either, need more info) |
This is why the second derivative appeared in Newton’s method: it tells us not only where the critical points are, but whether they are minima or maxima.
First vs Second Derivative: What Each Tells You
| First derivative \(f'(x)\) | Second derivative \(f''(x)\) | |
|---|---|---|
| Positive | Function is increasing | Function is concave up |
| Negative | Function is decreasing | Function is concave down |
| Zero | Candidate for max/min | Inconclusive (possible inflection) |
Combining both:
- \(f'(x) = 0\) and \(f''(x) > 0\): concave up at a flat point \(\Rightarrow\) local minimum.
- \(f'(x) = 0\) and \(f''(x) < 0\): concave down at a flat point \(\Rightarrow\) local maximum.