To find f'(1), we need to find the derivative of the function F(x) = (5x - 4)^6 with respect to x, and then evaluate it at x = 1.

Using the chain rule, we can find the derivative of F(x) as follows:

F'(x) = 6(5x - 4)^5 * 5
F'(x) = 30(5x - 4)^5

Now, to find f'(1), we substitute x = 1 into the derivative F'(x):

f'(1) = 30(5(1) - 4)^5
f'(1) = 30(5 - 4)^5
f'(1) = 30(1)^5
f'(1) = 30

Therefore, f'(1) = 30.