Differentiate each of the following, using whatever rules are necessary: (a) y = (4x27)(6x+5) 3.x (x =

Differentiate each of the following, using whatever rules are necessary:

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(a) y = (4x27)(6x+5) 3.x (x = 0) y' = Using the quotient rule and the product rule, 3x[(4x27)(6) + (6x+5) (8x)] - (4x - 7)(6x+5)(3) (3x)2 Simplifying algebraically, y': 3x [24x242 +48x2 +40x] - 3[24x +20x242x-35] 9.x2 144x360x2+105 y' = = 9x2 (b) f(x)=(9x+4) (2x-7)4 Using the product rule and the generalized power function rule, f' = (9x+4)[4(2x-7) (2)] + (2x-7)4 (9) Simplifying algebraically, 12x+7 (c) y = (5x+2)2 # - 5 2 f' = (9x+4) (8) (2x-7)3 + 9(2x-7)4 f' (72x+32) (2x-7)3 + 9(2x-7)4 = Using the quotient and generalized power function rules, dy == (5x+2)2(12) (12x+7)[2(5x+2)(5)] - Simplifying algebraically, (d) f(x)= 4x-5 3x+1 dy 313 dx = dx (5x+2)4 12(5x+2)2(12x+7)(50x +20) # 3 (5x+2) -300x2350x - 92 = (5x+2)4