The secant line through (x₀, f(x₀)) and (x₀+h, f(x₀+h)) has slope [f(x₀+h) − f(x₀)] / h. As h shrinks toward zero, that slope converges to the tangent — the exact derivative f′(x₀). This panel runs two derivative estimates side by side:
numeric (central diff): [f(x0+h) - f(x0-h)] / (2h)
analytic (exact rule): the calculus formula for f'
The central-difference estimate is deliberately evaluated at the current h so you can watch its error shrink roughly as O(h²) while h drops — and watch it degrade again if you push h below about 1e-4, where floating-point cancellation dominates. The second derivative f″(x₀), estimated the same way, tells you the concavity: positive curves up (convex), negative curves down (concave), near zero flags a possible inflection point.