Higher-Order Derivatives
When we differentiate a function once, we get the first derivative. But what happens when we differentiate again... and again... and again?
📐 Formal Definition
If $f(x)$ is a differentiable function, then its first derivative is $f'(x)$. If $f'(x)$ is itself differentiable, then its derivative is called the second derivative of $f$, written as $f''(x)$. In general, the $n$-th order derivative is obtained by differentiating $f(x)$ a total of $n$ times.
$$ f(x) \xrightarrow{\frac{d}{dx}} f'(x) \xrightarrow{\frac{d}{dx}} f''(x) \xrightarrow{\frac{d}{dx}} f'''(x) \xrightarrow{\frac{d}{dx}} f^{(4)}(x) \xrightarrow{\frac{d}{dx}} \cdots $$
✏️ Notation Guide
| Order | Lagrange | Leibniz |
|---|---|---|
| 1st | $f'(x)$ | $\frac{dy}{dx}$ |
| 2nd | $f''(x)$ | $\frac{d^2y}{dx^2}$ |
| 3rd | $f'''(x)$ | $\frac{d^3y}{dx^3}$ |
| $n$th | $f^{(n)}(x)$ | $\frac{d^ny}{dx^n}$ |
🌍 Physical Interpretation
Position: $s(t)$
Where an object is at time $t$
Velocity: $v(t) = s'(t)$
How fast position changes (1st derivative)
Acceleration: $a(t) = s''(t)$
How fast velocity changes (2nd derivative)
Key Insight: The second derivative $f''(x)$ tells us about the concavity of $f(x)$. If $f''(x) > 0$, the graph is concave up (smiles 😊). If $f''(x) < 0$, the graph is concave down (frowns 😞).