Question

a derivada da função f(x) ((3x+4))/(x^(6))

Answer 1

As derivadas de frações são dadas por     $\dfrac{u}{v} =\dfrac{u'v-uv'}{v^{2}}$  .

Passo a passo:

$f(x)=\dfrac{3x+4}{x^{6}} ~~,~~onde~u=3x+4~~e~~v=x^{6}\\ \\ \\ \dfrac{u'v-uv'}{v^{2}} ~~=~~\dfrac{\left[(3x+4)'~\cdot ~x^{6}\right]~-~\left[(3x+4)~\cdot~(x^{6})'\right]}{(x^{6})^{2}} ~~=~~\dfrac{\left[3\cdot x^{6}\right]-\left[(3x+4)\cdot 6x^{5}\rigth]}{x^{12}}$

$\dfrac{3x^{6}-(18x^{6}+24x^{5})}{x^{12}}~~=~~\dfrac{3x^{6}-18x^{6}-24x^{5}}{x^{12}}~~=~~\dfrac{-15x^{6}-24x^{5}}{x^{12}} \\ \\ \\ \\ \dfrac{-3x^{5}\cdot(5x+8)}{x^{12}} ~~=~~\boxed{-\dfrac{3\cdot(5x+8)}{x^{7}} }$

Resposta:

$\dfrac{\partial }{\partial x}\left(\dfrac{3x+4}{x^{6}} \right)~~=~~-\dfrac{3\cdot(5x+8)}{x^{7}}$

:)