derivative of 5^{2x^3}

解

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

e^{x^{3} l n (25)} \cdot 6 ln (5) x^{2}

解答ステップ

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

簡素化

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

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

指数の規則を適用する:

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

2 5^{x^{3}} = e^{x^{3} l n (25)}

= \frac{d}{d x} (e^{x^{3} l n (25)})

連鎖法則を適用:

e^{x^{3} l n (25)} \frac{d}{d x} (x^{3} ln (25))

= e^{x^{3} l n (25)} \frac{d}{d x} (x^{3} ln (25))

\frac{d}{d x} (x^{3} ln (25)) = 6 ln (5) x^{2}

= e^{x^{3} l n (25)} \cdot 6 ln (5) x^{2}