3. Calcule f ′(p), pela definição de derivada, sendo dados a) f(x) = x2 + x e p = 1 b) f(x) = 5x− 3 e p = −3 c) f(x) = √x e p = 3 d) f(x) = 2x3 −...

a) f(x) = x² + x e p = 1 Usando a definição de derivada, temos: f'(1) = lim (h → 0) [f(1 + h) - f(1)] / h f'(1) = lim (h → 0) [(1 + h)² + (1 + h) - (1² + 1)] / h f'(1) = lim (h → 0) [1 + 2h + h² + 1 + h - 2] / h f'(1) = lim (h → 0) [h² + 3h] / h f'(1) = lim (h → 0) [h(h + 3)] / h f'(1) = lim (h → 0) (h + 3) f'(1) = 3 Portanto, f'(1) = 3. b) f(x) = 5x - 3 e p = -3 Usando a definição de derivada, temos: f'(-3) = lim (h → 0) [f(-3 + h) - f(-3)] / h f'(-3) = lim (h → 0) [5(-3 + h) - 3 - (5(-3) - 3)] / h f'(-3) = lim (h → 0) [5h] / h f'(-3) = lim (h → 0) 5 f'(-3) = 5 Portanto, f'(-3) = 5. c) f(x) = √x e p = 3 Usando a definição de derivada, temos: f'(3) = lim (h → 0) [f(3 + h) - f(3)] / h f'(3) = lim (h → 0) [√(3 + h) - √3] / h Multiplicando o numerador e o denominador por √(3 + h) + √3, temos: f'(3) = lim (h → 0) [(3 + h) - 3] / [h(√(3 + h) + √3)] f'(3) = lim (h → 0) h / [h(√(3 + h) + √3)] f'(3) = lim (h → 0) 1 / (√(3 + h) + √3) f'(3) = 1 / (2√3) Portanto, f'(3) = 1 / (2√3). d) f(x) = 2x³ - x² e p = 1 Usando a definição de derivada, temos: f'(1) = lim (h → 0) [f(1 + h) - f(1)] / h f'(1) = lim (h → 0) [2(1 + h)³ - (1 + h)² - (2(1)³ - 1²)] / h f'(1) = lim (h → 0) [2(1 + 3h + 3h² + h³) - (1 + 2h + h²) - 7] / h f'(1) = lim (h → 0) [2 + 6h + 6h² + 2h³ - 1 - 2h - h² - 7] / h f'(1) = lim (h → 0) [2h³ + 5h² + 4h - 6] / h f'(1) = lim (h → 0) [h(2h² + 5h + 4) - 6h] / h f'(1) = lim (h → 0) 2h² + 5h - 6 f'(1) = 2(1)² + 5(1) - 6 f'(1) = 1 Portanto, f'(1) = 1. e) f(x) = 1/x e p = 1 Usando a definição de derivada, temos: f'(1) = lim (h → 0) [f(1 + h) - f(1)] / h f'(1) = lim (h → 0) [1 / (1 + h) - 1] / h f'(1) = lim (h → 0) [-h / (h(1 + h))] f'(1) = lim (h → 0) [-1 / (1 + h)] f'(1) = -1 Portanto, f'(1) = -1. f) f(x) = 1/x² e p = 2 Usando a definição de derivada, temos: f'(2) = lim (h → 0) [f(2 + h) - f(2)] / h f'(2) = lim (h → 0) [1 / (2 + h)² - 1 / 2²] / h f'(2) = lim (h → 0) [(1 / (4 + 4h + h²)) - (1 / 4)] / h f'(2) = lim (h → 0) [(1 / (4 + 4h + h²)) - (4 / 16)] / h f'(2) = lim (h → 0) [(1 / (4 + 4h + h²)) - (1 / 4)] / h Multiplicando o numerador e o denominador por (4 + 4h + h²), temos: f'(2) = lim (h → 0) [1 - (4 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-3 - 4h - h²] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0) [-(3 + 4h + h²)] / [4h(4 + 4h + h²)] f'(2) = lim (h → 0)