[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 + 2v^1/2] + [v³ - v^1/2][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