help with my questions for math

1) Differentiate each of the following functions. Final answers should not contain negative or fraction exponents and should be simplied/factored as much as possible. a) f (x) : m (K 3 marks) 1)) f(x) : #5 (K 3 marks) 2 (4x2 + 2x 7) 0) f(x) = (6x2 5)5(2x 1)'1 (K 7 3 marks) d) f(x) = (52:34 (K 3 marks) x i 2 e) f (x ) = V4x + 5 (x 2 - 3) (A - 3 marks) f) f (x ) = 3- x2 + 1 3x - 4 (A - 3 marks) 2) Find the derivative of f(x) =-7x2 +5x -8 using first principles. (A -3 marks)3) The owners of a restaurant are interested analyzing the productivity of their staff. The function NU) =150 600 6+3:2 staff after 1' hours during an 8-hour workday (0 S t S 8) . Determine the rate, to the nearest tenth, at which the customers are being served at 4 hours. (A , 3 marks) models the total number of customers, N, served by the 4) Determine the equation of the tangent to f(x) = x2 + 3x +1 that is perpendicular to the line y=x+2. (176marks) 5) Find numbers a, b, and c such that the graph of y = ax2 + bx+ c has xintercepts of 0 and 8, and a tangent with a slope of 16 at x : 2. (1 6 marks) (more space available on next page) 6) Fill in the boxes below with integers to create the equation of another function that has the same tangent slopes as y : 78x3 + 4x2 , 7x + 5 at all x-values. (C 2 marks) 7) Prove the Product Rule of differentiation. That is, prove that if p(x) = f (x)g(x) , where f(x) and g(x) are differentiable, then p'(x) = _f"(x)g(x)+ g'(x)f(x). (Ci 4 marks)