Question

Let g(2x) = x h(x²) And g(x) = f(x²)Determine an expression for h'(x²) in terms of f and f'​

Answer 1

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)