The derivative the function f(x) = 8x^2 + 11x at x = 7 can be determined by taking the derivative of each term in the function first through differential calculus. Using power rule where power decreases by one and the power is multiplied to the numerical coeff. In this case, f'(x) = 16x + 11. Substituting x = 7, f'(x) = 16*7 + 11 equal to 123.