Question

Calcule a derivada f(x)= 1/(x^5), pela regra de derivação

Answer 1

Derivada - Regra do quociente :

$\displaystyle [\frac{\text f }{\text g}]' = \frac{\text f'.\text g-\text f.\text g' }{\text g^2}$

bora

$\displaystyle \text{f(x)} = \frac{1}{\text x^5}$

derivando pela regra do quociente :

$\displaystyle \text{f'(x)} = [\frac{1}{\text x^5}]' \to \text{f'(x)} = \frac{1'.\text x^5-1.(\text x^5)'}{(\text x^5)^2}$

$\displaystyle \text{f'(x)} = \frac{0.\text x^5-1.5\text x^4}{\text x^{10}}$

$\huge\boxed{\displaystyle \text{f'(x)} = \frac{-5}{\text x^{6}}}$

A forma mais rápida é reescrevendo assim :

$\displaystyle \text{f(x)} = \frac{1}{\text x^5} \to \boxed{\text{f(x)} = \text x^{-5}}$

derivando :

$\text{f'(x)} = [x^{-5}]' \to \text{f'(x)} = -5.\text x^{-5-1}$

$\text{f'(x)} = -5.\text x^{-6}$

$\boxed{\displaystyle \text{f'(x)} = \frac{-5}{\text x^6}}$