derivative (1+4/x)^x

解

derivative

(ln (1 + \frac{4}{x}) - \frac{4}{x + 4}) (1 + \frac{4}{x})^{x}

解答ステップ

\frac{d}{d x} ((1 + \frac{4}{x})^{x})

指数の規則を適用する:

a^{b} = e^{b l n (a)}

(1 + \frac{4}{x})^{x} = e^{x l n (1 + \frac{4}{x})}

= \frac{d}{d x} (e^{x l n (1 + \frac{4}{x})})

連鎖法則を適用:

e^{x l n (1 + \frac{4}{x})} \frac{d}{d x} (x ln (1 + \frac{4}{x}))

= e^{x l n (1 + \frac{4}{x})} \frac{d}{d x} (x ln (1 + \frac{4}{x}))

\frac{d}{d x} (x ln (1 + \frac{4}{x})) = ln (1 + \frac{4}{x}) - \frac{4}{x + 4}

= e^{x l n (1 + \frac{4}{x})} (ln (1 + \frac{4}{x}) - \frac{4}{x + 4})

e^{x l n (1 + \frac{4}{x})} = (1 + \frac{4}{x})^{x}

= (ln (1 + \frac{4}{x}) - \frac{4}{x + 4}) (1 + \frac{4}{x})^{x}