Lesson 10: Higher-Order Derivatives & Acceleration

1. Second Derivative – Concavity & Acceleration

The first derivative \(f'(x)\) = slope = velocity (if position function).

The second derivative \(f''(x)\) = rate of change of slope = concavity / acceleration.

• \(f''(x) > 0\) → concave up (smiles), acceleration positive
• \(f''(x) < 0\) → concave down (frowns), acceleration negative
• \(f''(x) = 0\) → possible inflection point (change in concavity)

Example: \(s(t) = t^3 - 3t^2\) (position)

\(v(t) = s'(t) = 3t^2 - 6t\)

\(a(t) = s''(t) = 6t - 6\)

2. Third & Higher Derivatives

Third derivative: rate of change of acceleration (jerk in physics)

Example: position \(s(t) = t^4\)

• velocity \(v = 4t^3\)
• acceleration \(a = 12t^2\)
• jerk \(j = 24t\)

In ML: higher derivatives appear in Hessian (second-order optimization), Taylor series.