Numeric Function Derivative
Computes numeric derivative of a math expression at a point using central difference method.
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Numerical derivative of a function
A numerical derivative estimates f'(x) from sampled function values, which is what you reach for when there's no analytic form or it costs too much to compute. The forward difference f'(x) ≈ (f(x+h) − f(x))/h has error O(h). The central difference f'(x) ≈ (f(x+h) − f(x−h))/(2h) brings that down to O(h²), and a 5-point stencil (−f(x+2h)+8f(x+h)−8f(x−h)+f(x−2h))/(12h) reaches O(h⁴). Take f(x) = x² at x = 2 with h = 0.01: central gives (4.0401 − 3.9601)/0.02 = 4.0000, which matches the exact 2x = 4. Pick h around √ε for forward or ∛ε for central, with ε being machine epsilon. Go too small and subtractive cancellation wrecks your precision.
Applications and context
You'll find numerical derivatives behind finite-element analysis (FEM), numerical ODE/PDE solvers, optimization without analytic gradients, signal processing (rate-of-change detection) and economic sensitivity studies. In ML the story is different. Automatic differentiation (PyTorch autograd, JAX, TensorFlow) usually wins over finite differences, since it hands back gradients exact up to round-off and scales to millions of parameters.
FAQ
Forward vs central, which one? Go central whenever you can evaluate both sides of x. That one extra call buys you a whole order of accuracy.
Can I just use h = 10⁻¹⁵? No. When you subtract two near-equal floats you throw away significant digits, and the total error traces a U-shape with one optimal h sitting at the bottom.
How does autodiff differ? It walks the chain rule symbolically through your source code, so you get exact derivatives and never have to tune an h.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.