[derivatives - rational exponents][section 8]

(1.) f(x) = x^2/5 + 2x^1/3

f'(x) = (2/5)x^-3/5 + (2/3)x^-2/3 use the power rule

(2.) f(x) = 5x^-2/9

f'(x) = (-10/9)x^-11/9 use the power rule

(3.) f(x) = (x² + 1)^5/3

f'(x) = (10x/3)(x² + 1)^2/3 use the chain rule

(4.) f(x) = (x² - 1)^-2/3

f'(x) = (-4x/3)(x² - 1)^-5/3 use the chain rule

(5.) f(x) = (x³ + 3)^1/10

f'(x) = (3x²/10)(x³ + 3)^-9/10 use the chain rule

(6.) f(x) = (x⁵ + 2x + 1)^3/2

f'(x) = (3/2)(5x⁴ + 2)(x⁵ + 2x + 1)^1/2 use the chain rule

(7.) s(t) = (t¹⁰ - 2)^4/7

s'(t) = (4/7)(10t⁹)(t¹⁰ - 2)^-3/7 use the chain rule

(8.) s(t) = (t³ + t² + 1)^5/8

s'(t) = (5/8)(3t² + 2t)(t³ + t² + 1)^-3/8 use the chain rule
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(9.) g(u) = \³/u³ + 3u + 1

g(u) = (u³ + 3u + 1)^1/3 write using the 1/3 power

g'(u) = (1/3)(3u² + 3)(u³ + 3u + 1)^-2/3 use the chain rule

g'(u) = (u² + 1)(u³ + 3u + 1)^-2/3 multiply thru
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(10.) s(t) = (t² + 1)^√3
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s'(t) = (2t√3)(t² + 1)^{(√3 - 1)} use the chain rule