derivative of (6x+5^x)

解

\frac{d}{d x} ((6 x + 5)^{x})

(ln (6 x + 5) + \frac{6 x}{6 x + 5}) (6 x + 5)^{x}

解答ステップ

\frac{d}{d x} ((6 x + 5)^{x})

指数の規則を適用する:

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

(6 x + 5)^{x} = e^{x l n (6 x + 5)}

= \frac{d}{d x} (e^{x l n (6 x + 5)})

連鎖法則を適用:

e^{x l n (6 x + 5)} \frac{d}{d x} (x ln (6 x + 5))

= e^{x l n (6 x + 5)} \frac{d}{d x} (x ln (6 x + 5))

\frac{d}{d x} (x ln (6 x + 5)) = ln (6 x + 5) + \frac{6 x}{6 x + 5}

= e^{x l n (6 x + 5)} (ln (6 x + 5) + \frac{6 x}{6 x + 5})

e^{x l n (6 x + 5)} = (6 x + 5)^{x}

= (ln (6 x + 5) + \frac{6 x}{6 x + 5}) (6 x + 5)^{x}