Question

Utilizando a definição determine a derivada da função f(x) = 2 – x³, no ponto – 2.

Answer 1

Resposta:

Explicação passo-a-passo:

$\displaystyle \frac{\partial f(x)}{\partial x} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\\\\\ \frac{\partial f(x)}{\partial x} = \lim_{h \to 0} \frac{2-(x+h)^3-(2-x^3)}{h} = \lim_{h \to 0} \frac{\diagup\!\!\!\!2-(x^3+h^3+3x^2h+3xh^2)-\diagup\!\!\!\!2+x^3}{h}=\\\\\\\lim_{h \to 0} \frac{-\diagup\!\!\!\!x^3-h^3-3x^2h-3xh^2+\diagup\!\!\!\!x^3}{h}=\lim_{h \to 0} \frac{-\diagup\!\!\!\!h(h^2+3x^2+3xh)}{\diagup\!\!\!\!h}=-3x^2\\\\\\\frac{\partial f(-2)}{\partial x} =-3(-2)^2=-3.4=-12$

obs.

$\displaystyle (a+b)^3=a^3+b^3+3a^2b+3ab^2$