derivative of 3^{x^4-2x^2}

解

\frac{d}{d x} (3^{x^{4} - 2 x^{2}})

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

解答ステップ

\frac{d}{d x} (3^{x^{4} - 2 x^{2}})

指数の規則を適用する:

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

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

= \frac{d}{d x} (e^{(x^{4} - 2 x^{2}) l n (3)})

連鎖法則を適用:

e^{(x^{4} - 2 x^{2}) l n (3)} \frac{d}{d x} ((x^{4} - 2 x^{2}) ln (3))

= e^{(x^{4} - 2 x^{2}) l n (3)} \frac{d}{d x} ((x^{4} - 2 x^{2}) ln (3))

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

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

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

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