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Exercício 1 a) f(x) = √x f'(x0) = lim f(x) - f(x0) f'(x0) = lim √x - √9 f'(x0) = lim √x - 3 x0 = 9 x - x0 x - 9 x - 9 x →x0 x →9 x →9 f'(x0) = lim (√x - 3).(√x + 3) f'(x0) = lim (√x - 3).(√x + 3) f'(x0) = lim √x² - 3² (x - 9).(√x + 3) (x - 9).(√x + 3) (x - 9).(√x + 3) x →9 x →9 x →9 f'(x0) = lim x - 9 f'(x0) = lim 1 f'(x0) = lim 1 (x - 9).(√x + 3) √x + 3 .6 x →9 x →9 x →9 b) f(x) = x⁵ - 5x³ + 5x - 7 x0 = 1 f'(x0) = lim x⁵ - 5x³ + 5x - 7 - x0⁵ + 5x0³ - 5x0 + 7 x - 1 x → x0 f'(x0) = lim (x⁵ - x0⁵) . 5(x³-x0³) + 5(x-x0) x - x0 x → 1 f'(x0) = lim (x⁴ + x³x0 + x²x0² + xx0³ + x0⁴) x → 1 f'(x0) = lim 5x0⁴ - 53x² _x0² + xx0³ + x0⁴) x → 1 f'(x0) = lim 5x0⁴ - 15x0 +5 x → 1 f'(x0) = lim -5 x → x0 c) f(x) = x³ - 4x + 5 x0 = 4 x - 5 f'(4) = lim x³ - 4x + 5 + 53 f'(4) = lim x³ - 4x + 5 + 53x - 265 f'(4) = lim x³ + 49x -260 x - 5 (x - 4) (x - 5) (x - 4) (x - 5) x - 4 x → 4 x → 4 x → 4 f'(4) = lim (x - 4) x² + 4x + 65 f'(4) = lim x² + 4x + 65 f'(4) = lim - 97 (x - 4) (x - 5) (x - 5) x → 4 x → 4 x → 4
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