The Derivative

The Derivative

The derivative of $f$ at $x$ is the instantaneous rate of change — the slope of the tangent line at that point:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Central Difference Formula

The forward difference $\frac{f(x+h) - f(x)}{h}$ has error $O(h)$. The central difference is more accurate — error $O(h^2)$:

$f'(x) \approx \frac{f(x+h) - f(x-h)}{2h}$

This is what you should implement. With h = 1e-6, results match analytic derivatives to 9+ significant figures.

Analytic Derivatives

For reference, the exact rules you can verify numerically:

Verification

For $f(x) = x^2$:

• Analytic: $f'(2) = 2 \cdot 2 = 4$
• Central difference: $\frac{(2+h)^2 - (2-h)^2}{2h} = \frac{8h}{2h} = 4$ ✓ (exact, regardless of $h$)

Your Task

Implement double derivative(double (*f)(double), double x, double h) using the central difference formula.