derivative of 3^{sin(3x})

解

\frac{d}{d x} (3^{s i n (3 x)})

ln (3) \cdot 3^{s i n (3 x) + 1} cos (3 x)

解答ステップ

\frac{d}{d x} (3^{s i n (3 x)})

指数の規則を適用する:

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

3^{s i n (3 x)} = e^{s i n (3 x) l n (3)}

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

連鎖法則を適用:

e^{s i n (3 x) l n (3)} \frac{d}{d x} (sin (3 x) ln (3))

= e^{s i n (3 x) l n (3)} \frac{d}{d x} (sin (3 x) ln (3))

\frac{d}{d x} (sin (3 x) ln (3)) = ln (3) cos (3 x) \cdot 3

= e^{s i n (3 x) l n (3)} ln (3) cos (3 x) \cdot 3

簡素化

e^{s i n (3 x) l n (3)} ln (3) cos (3 x) \cdot 3 : ln (3) \cdot 3^{s i n (3 x) + 1} cos (3 x)

= ln (3) \cdot 3^{s i n (3 x) + 1} cos (3 x)