[the product rule][section 5]

(1.)   f(x) = (x2 - 9)(x - 5)    here is the problem

f'(x) = 2x(x - 5) + 1(x2 - 9)  use the product rule

f'(x) = 2x2 - 10x + x2 - 9    multiply thru

f'(x) = 3x2 - 10x - 9        combine like terms

(2.)  g(t) = (9 - t2)(5 - t)       here is the problem

g'(t) = -2t(5 - t) - 1(9 - t2)  use the product rule

g'(t) = -10t + 2t2 - 9 + t2    multiply thru

g'(t) = 3t2 - 10t - 9         combine like terms

(3.)  f(x) = 2x(x2 + 1)             here is the problem

f'(x) = 2(x2 + 1) + 2x(2x)  use the product rule

f'(x) = 2x2 + 2 + 4x2            multiply

f'(x) = 6x2 + 2         combine like terms

(4.)   F(x) = (1 + x + x5)(2 - x + x6)   here is the problem

F'(x) = (1 + 5x4)(2 - x + x6) + (-1 + 6x5)(1 + x + x5)

[use the product rule]
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(5.)   g(t) = t3(t + √t)           here is the problem
_
g'(t) = 3t2(t + √t) + [1 + (1/2)t-1/2](t3)

[use the product rule]
_       _
(6.)  g(t) = (1 + √t)(1 - √t)     here is the problem

g(t) = 1 - t   foil multiply combine like terms

g'(t) = -1    use the power rule
_        __
(7.)  p(v) = (v3 - √v)(v2 + 2√v)        here is the problem

p(v) = [3v2 - (1/2)v-1/2][v2 + 2v1/2] + [v3 - v1/2][2v + v-1/2]

[use the product rule]

(8.)  v(x) = (1 - x-2)(1 - x2)         here is the problem

v'(x) = (2x-3)(1 - x2) - (2x)(1 - x-2)

[use the product rule]

(9.)  v(x) = x(1 - x)             here is the problem

v(x) = x - x2          multiply thru

v'(x) = 1 - 2x   use the power rule

(10.)  f(x) = 4x(x5 + 1)      here is the problem

f'(x) = 4(x5 + 1) + (5x4)(4x)   use the product rule

f'(x) = 4x5 + 4 + 20x5            multiply thru

f'(x) = 24x5 + 5                combine like terms