derivative h(x)=(4x+6)^x

解

導関数

h (x) = (4 x + 6)^{x}

(ln (4 x + 6) + \frac{2 x}{2 x + 3}) (4 x + 6)^{x}

解答ステップ

\frac{d}{d x} ((4 x + 6)^{x})

指数の規則を適用する:

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

(4 x + 6)^{x} = e^{x l n (4 x + 6)}

= \frac{d}{d x} (e^{x l n (4 x + 6)})

連鎖法則を適用:

e^{x l n (4 x + 6)} \frac{d}{d x} (x ln (4 x + 6))

= e^{x l n (4 x + 6)} \frac{d}{d x} (x ln (4 x + 6))

\frac{d}{d x} (x ln (4 x + 6)) = ln (4 x + 6) + \frac{2 x}{2 x + 3}

= e^{x l n (4 x + 6)} (ln (4 x + 6) + \frac{2 x}{2 x + 3})

e^{x l n (4 x + 6)} = (4 x + 6)^{x}

= (ln (4 x + 6) + \frac{2 x}{2 x + 3}) (4 x + 6)^{x}