Question

Derivada de
$y = 5 {}^{ - \frac{1}{x} }$

Answer 1

Olá.

Nesse caso de função exponencial, teremos que aplicar propriedades de logaritmos naturais para tentar encontrar a derivada:

$\displaystyle y = 5^{-\frac{1}{x}} \\\\ \ln y = \ln 5^{-\frac{1}{x}} \\\\ \ln y = -\frac{1}{x} \cdot \ln 5 \\\\ \ln y = -\ln 5 \cdot \frac{1}{x}$

Derivando ambos os lados implicitamente nós obtemos:

$\displaystyle \frac{1}{y} \frac{dy}{dx} = - \ln 5 \cdot \bigg(-\frac{1}{x^2} \bigg) \\\\\\ \frac{1}{y} \frac{dy}{dx} = \frac{\ln 5}{x^2} \\\\\\ \frac{dy}{dx} = y \cdot \frac{\ln 5}{x^2} \\\\\\ \frac{dy}{dx} = 5^{-\frac{1}{x}} \cdot \frac{\ln 5}{x^2} \\\\\\ \frac{dy}{dx} = \frac{1}{5^{\frac{1}{x}}} \cdot \frac{\ln 5}{x^2} \\\\\\ \frac{dy}{dx} = \frac{1}{ \sqrt[x]{5} } \cdot \frac{\ln 5}{x^2} \\\\\\ \frac{dy}{dx} = \frac{\ln 5}{x^2 \sqrt[x]{5}}$