derivative 5^{3^{x^2}}

解

導関数

5^{3^{x^{2}}}

2 ln (5) ln (3) \cdot 5^{3^{x^{2}}} \cdot 3^{x^{2}} x

解答ステップ

\frac{d}{d x} (5^{3^{x^{2}}})

指数の規則を適用する:

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

5^{3^{x^{2}}} = e^{3^{x^{2}} l n (5)}

= \frac{d}{d x} (e^{3^{x^{2}} l n (5)})

連鎖法則を適用:

e^{3^{x^{2}} l n (5)} \frac{d}{d x} (3^{x^{2}} ln (5))

= e^{3^{x^{2}} l n (5)} \frac{d}{d x} (3^{x^{2}} ln (5))

\frac{d}{d x} (3^{x^{2}} ln (5)) = 2 ln (5) ln (3) \cdot 3^{x^{2}} x

= e^{3^{x^{2}} l n (5)} \cdot 2 ln (5) ln (3) \cdot 3^{x^{2}} x

e^{3^{x^{2}} l n (5)} = 5^{3^{x^{2}}}

= 2 ln (5) ln (3) \cdot 5^{3^{x^{2}}} \cdot 3^{x^{2}} x