Question

calcule as derivadas da função abaixo.
$f{x} = \frac{1}{x} + \frac{1}{ {x}^{2} } + \frac{x}{3}$
$f{x} = - \frac{ { - x}^{2} }{16} - {x}^{ \frac{2}{3} } + \frac{1}{3 {x}^{2} } + \frac{x}{3}$
passo a passo please.​

Answer 1

 a -

$f(x)=\frac{1}{x}+\frac{1}{x^2}+\frac{x}{3}\\\\f'(x)=\frac{-1}{x^2}+\frac{-2.x}{x^4}+\frac{3}{9}\\\\f'(x)=-\frac{1}{x^2}-\frac{2}{x^3}+\frac{1}{3}$

 b -

$f(x)=\frac{x^2}{16}-x^{\frac{2}{3}}+\frac{1}{3.x^2}+\frac{x}{3}\\\\f'(x)=\frac{2.x.16}{256}-\frac{2}{3}.x^{-\frac{1}{3}}+\frac{-6.x}{9.x^4}+\frac{3}{9}\\\\f'(x)=\frac{2.x}{16}-\frac{2}{3.x^{\frac{1}{3}}}-\frac{2}{3.x^3}+\frac{1}{3}\\\\f'(x)=\frac{x}{8}-\frac{2}{3.\sqrt[3]{x} }-\frac{2}{3.x^3}+\frac{1}{3}$

 Propriedades utilizadas:

$f(x)=g(x)+h(x)\\f'(x)=g'(x)+h'(x)$

$f(x)=\frac{g(x)}{h(x)}\\f'(x)=\frac{g'(x).h(x)-g(x).h'(x)}{[h(x)]^2}$

$f(x)=x^n\\f'(x)=n.x^{n-1}$

Dúvidas só perguntar!