derivative calculation

d(f(4x^2))/dx = 9x^2

[d(f(4x^2))/d(4x^2)][d(4x^2)/dx] = 9x^2

[d(f(4x^2))/d(4x^2)][8x] = 9x^2

[d(f(4x^2))/d(4x^2)] = (9/8)x

Let y = 4x^2

d(f(y))/dy = (9/8)[(√y)/2]

= (9√y)/16

But x or y are just symbols. therefore

f'(x) = (9√x)/16

d/dx(f(4x^2))=9x^2

integrating both sides

f(4x^2) = 3x^3 + C where C is a constant

let y = 4x^2 => x = (y/4)^(1/2)

f(y) = 3(y/4)^(3/2) + C

f'(x) = (9/8)(x/4)^(1/2)