derivative of 3^{sin(pix})

解

\frac{d}{d x} (3^{s i n (π x)})

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

解答ステップ

\frac{d}{d x} (3^{s i n (π x)})

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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