Numerical Differentiation

Numerical Analysis · Axiom Academy

INTRO Numerical Differentiation What if you know the function values but need the slope? Remember the definition of a derivative? It's the limit of a slope as two points get infinitely close. But what if we can't take the limit? Explore these scenarios where analytical derivatives aren't available: Temperature readings, stock prices, sensor measurements... You have discrete data points, not a formula. How do you find the rate of change? The simplest approach: look at the slope from the current point to the next point. ↔ The Central Difference: Smarter Approximation What if we use points on both sides of x? Watch how much better this works! Numerical differentiation approximates derivatives using function values at discrete points, bridging the gap between calculus theory and computational practice. Forward: f'(x) ≈ [f(x+h) - f(x)] / h Backward: f'(x) ≈ [f(x) - f(x-h)] / h Central: f'(x) ≈ [f(x+h) - f(x-h)] / (2h) Smaller h means better approximation but more roundoff error. Finding the sweet spot is the art of numerical differentiation!

This is the written version of the interactive lesson above. See the full Numerical Analysis course.