Question

Calcule f’(x), f’’(x) f’’’(x) nas funções abaixo:

f(x) = 2x^6+x^3

f(x) = 1/x

Answer 1

Resposta:

Explicação passo a passo:

$\text{Usando a regra: } \, \, f(x) = a.x^n \,\, \to \,\, f'(x) = a.n.x^{n-1}\\\\\text{Logo, temos que: } f'(x) = 2. 6 . x^{5} + 3.x^2 = 12. x^{5} + 3.x^2 \,\, \to f'(x) =12. x^{5} + 3.x^2 \\\\f"(x) = 12. 5. x^{4} + 3.2.x = 60 . x^{4} + 6.x \,\, \to \,\, f"(x) = 60 . x^{4} + 6.x\\\\f''' (x) = 60 . 4 . x ^{3} + 6 = 240 . x ^{3} + 6 \,\, \to \,\, f''' (x) = 240 . x ^{3} + 6$

Para o segundo item:

$\text{Temos que: } \,\, f(x) = \dfrac{1}{x} = x^{-1} \,\, \to \,\, f(x) = x^{-1} \\\\\tex{Derivando, temos que: } \,\, f'(x) = (-1).x^{-2} \,\, \to \,\, f'(x) = -x^{-2}\\\\f"(x) = - (-2) . x^{-3} = 2.x^{-3} \,\, \to \,\, f"(x) = 2.x^{-3}\\\\f'''(x) = (-3).(2).x^{-4} = - 6..x^{-4} \,\, \to \,\, f'''(x) = - 6..x^{-4}$