derivative of 3^{2x^2}

解

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

4 ln (3) \cdot 9^{x^{2}} x

解答ステップ

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

簡素化

3^{2 x^{2}} : 9^{x^{2}}

= \frac{d}{d x} (9^{x^{2}})

指数の規則を適用する:

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

9^{x^{2}} = e^{x^{2} l n (9)}

= \frac{d}{d x} (e^{x^{2} l n (9)})

連鎖法則を適用:

e^{x^{2} l n (9)} \frac{d}{d x} (x^{2} ln (9))

= e^{x^{2} l n (9)} \frac{d}{d x} (x^{2} ln (9))

\frac{d}{d x} (x^{2} ln (9)) = 4 ln (3) x

= e^{x^{2} l n (9)} \cdot 4 ln (3) x

e^{x^{2} l n (9)} = 9^{x^{2}}

= 4 ln (3) \cdot 9^{x^{2}} x