SOLUTION: The gradient/slope of a function of f(x) = (2x-b)^a is f'(x) = 24x^2-24x+6, find the constant a and b. I'm a confused on how to start on this.

The wording of the problem means that the derivative of the function f(x) = (2x-b)^a

is the polynomial of degree 2,  f'(x) = 24x^2 - 24x + 6.



It means that f(x) is a polynomial of degree 3;  hence, a= 3.



Next, you can make FOIL  of  (2x-b)^3  to get  (2x-b)^3 = 8x^3 - 3*(2x^2)*b + . . . = 8x^2 - 12x^2*b + . . . 



Then, looking at its term of degree 2,  -3*(2x^2)*b = -3*(4x^2)*b = -12x^2*b,  and comparing it 

with its derivative -24x, you conclude that b= 1.



So,  the constants "a" and "b" are  a= 3,  b= 1.    


Thus  f(x) = (2x-1)^3 = 8x^3 - 3*(2x)^2*1 + 3*(2x)*1^2 - 1 = 8x^3 - 12x^2 + 6x - 1.


ANSWER.  a= 3, b= 1.