The Second Derivative, Convexity, and Inflection Points
A complete guide to higher-order derivatives, concavity analysis, and function investigation
1. The Second Derivative — Definition, Notation, and Physical Meaning
1.1 Definition
The second derivative of a function $f(x)$ is simply the derivative of the first derivative. If $f'(x)$ exists and is itself differentiable, then:
$$f''(x) = \frac{d}{dx}\left(\frac{df}{dx}\right) = \frac{d^2 f}{dx^2}$$
1.2 Notation
Several equivalent notations are used across textbooks:
| Notation | Name / Context |
|---|---|
| $f''(x)$ | Lagrange notation (most common in algebra/calculus courses) |
| $y''$ | Shorthand when $y = f(x)$ |
| $\dfrac{d^2 y}{dx^2}$ | Leibniz notation (emphasises the variable) |
| $D^2 f$ | Operator notation |
1.3 Mechanical Computation
To find $f''(x)$, simply differentiate $f'(x)$ using the same rules (power rule, product rule, chain rule, etc.).
Quick example: Let $f(x) = x^4 - 6x^2 + 5$.
Step 1: $f'(x) = 4x^3 - 12x$
Step 2: $f''(x) = 12x^2 - 12$
Step 1: $f'(x) = 4x^3 - 12x$
Step 2: $f''(x) = 12x^2 - 12$
1.4 Physical Interpretation — Acceleration
If $s(t)$ represents the position of an object at time $t$, then:
- $s'(t) = v(t)$ is the velocity (rate of change of position)
- $s''(t) = v'(t) = a(t)$ is the acceleration (rate of change of velocity)
Speeding up vs. slowing down:
- If $v(t)$ and $a(t)$ have the same sign → the object is speeding up
- If $v(t)$ and $a(t)$ have opposite signs → the object is slowing down
Example: A ball thrown upward: $s(t) = -4.9t^2 + 9.8t$ (metres).
$v(t) = s'(t) = -9.8t + 9.8$
$a(t) = s''(t) = -9.8$ m/s²
The acceleration is constant and negative (gravity pulls downward). When $t < 1$, $v(t) > 0$ (moving up) and $a < 0$ → slowing down. When $t > 1$, $v(t) < 0$ (moving down) and $a < 0$ → speeding up.
$v(t) = s'(t) = -9.8t + 9.8$
$a(t) = s''(t) = -9.8$ m/s²
The acceleration is constant and negative (gravity pulls downward). When $t < 1$, $v(t) > 0$ (moving up) and $a < 0$ → slowing down. When $t > 1$, $v(t) < 0$ (moving down) and $a < 0$ → speeding up.