derivative (x+3)^{3x-2}

解

導関数

(x + 3)^{3 x - 2}

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

解答ステップ

\frac{d}{d x} ((x + 3)^{3 x - 2})

指数の規則を適用する:

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

(x + 3)^{3 x - 2} = e^{(3 x - 2) l n (x + 3)}

= \frac{d}{d x} (e^{(3 x - 2) l n (x + 3)})

連鎖法則を適用:

e^{(3 x - 2) l n (x + 3)} \frac{d}{d x} ((3 x - 2) ln (x + 3))

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

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

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

e^{(3 x - 2) l n (x + 3)} = (x + 3)^{3 x - 2}

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