To find the derivative of the given function ( f(x) = 4x^3 + 12x^2 - 3 ), we need to use the power rule for differentiation.

Given function: ( f(x) = 4x^3 + 12x^2 - 3 )

To find the derivative of ( f(x) ), denoted as ( f'(x) ), we differentiate each term of the function with respect to ( x ):

( f'(x) = 4(3)x^(3-1) + 12(2)x^(2-1) - 0 )
( f'(x) = 12x^2 + 24x )

Now, to find the derivative value at ( x = 1 ), we substitute ( x = 1 ) in the derivative function:

( f'(1) = 12(1)^2 + 24(1) )
( f'(1) = 12 + 24 )
( f'(1) = 36 )

Therefore, the value of the derivative of ( f(x) = 4x^3 + 12x^2 - 3 ) at ( x = 1 ) is ( 36 ).