derivative f(x)=3^{2x^2+3x-1}

解

導関数

f (x) = 3^{2 x^{2} + 3 x - 1}

ln (3) \cdot 3^{2 x^{2} + 3 x - 1} (4 x + 3)

解答ステップ

\frac{d}{d x} (3^{2 x^{2} + 3 x - 1})

指数の規則を適用する:

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

3^{2 x^{2} + 3 x - 1} = e^{(2 x^{2} + 3 x - 1) l n (3)}

= \frac{d}{d x} (e^{(2 x^{2} + 3 x - 1) l n (3)})

連鎖法則を適用:

e^{(2 x^{2} + 3 x - 1) l n (3)} \frac{d}{d x} ((2 x^{2} + 3 x - 1) ln (3))

= e^{(2 x^{2} + 3 x - 1) l n (3)} \frac{d}{d x} ((2 x^{2} + 3 x - 1) ln (3))

\frac{d}{d x} ((2 x^{2} + 3 x - 1) ln (3)) = ln (3) (4 x + 3)

= e^{(2 x^{2} + 3 x - 1) l n (3)} ln (3) (4 x + 3)

e^{(2 x^{2} + 3 x - 1) l n (3)} = 3^{2 x^{2} + 3 x - 1}

= ln (3) \cdot 3^{2 x^{2} + 3 x - 1} (4 x + 3)