Differentiating both sides with the product and chain rules gives
2 g'(2x) = h(x²) + 2x² h'(x²)
Solving for h'(x²) gives
h'(x²) = (2 g'(2x) - h(x²))/(2x²)
If g(x) = f(x²), then g'(x) = 2x f'(x²). Making the appropriate replacements, we get
h'(x²) = (2 g'(2x) - h(x²))/(2x²)
h'(x²) = (2 g'(2x) - g(2x)/x)/(2x²)
h'(x²) = (2 • 2 (2x) f'((2x)²) - f((2x)²)/x)/(2x²)
h'(x²) = (8x f'(4x²) - f(4x²)/x)/(2x²)
h'(x²) = (8x² f'(4x²) - f(4x²))/(2x³)
