Chain Rule

The Chain Rule

The chain rule handles derivatives of composed functions $f(g(x))$:

$\frac{d}{dx}\bigl[f(g(x))\bigr] = f'(g(x)) \cdot g'(x)$

Examples

Numerical Chain Derivative

We can approximate the chain derivative numerically using the central difference formula applied to the composition:

$\frac{d}{dx}\bigl[f(g(x))\bigr] \approx \frac{f(g(x+h)) - f(g(x-h))}{2h}$

This is just the standard central difference, but applied to the composed function $f \circ g$:

double chain_deriv(double (*f)(double), double (*g)(double),
                   double x, double h) {
    return (f(g(x + h)) - f(g(x - h))) / (2.0 * h);
}

Verification

For $h(x) = \sin(x^2)$ at $x = 1$:

• Analytic: $h'(1) = \cos(1^2) \cdot 2 \cdot 1 = 2\cos(1) \approx 1.0806$
• Numerical with $h = 10^{-6}$: should match to 4+ decimal places

Your Task

Implement double chain_deriv(f, g, x, h) using the central difference on the composition $f(g(x))$.