derivative of 5^{xsin(x})

解

\frac{d}{d x} (5^{x s i n (x)})

ln (5) \cdot 5^{x s i n (x)} (sin (x) + x cos (x))

解答ステップ

\frac{d}{d x} (5^{x s i n (x)})

指数の規則を適用する:

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

5^{x s i n (x)} = e^{x s i n (x) l n (5)}

= \frac{d}{d x} (e^{x s i n (x) l n (5)})

連鎖法則を適用:

e^{x s i n (x) l n (5)} \frac{d}{d x} (x sin (x) ln (5))

= e^{x s i n (x) l n (5)} \frac{d}{d x} (x sin (x) ln (5))

\frac{d}{d x} (x sin (x) ln (5)) = ln (5) (sin (x) + x cos (x))

= e^{x s i n (x) l n (5)} ln (5) (sin (x) + x cos (x))

e^{x s i n (x) l n (5)} = 5^{x s i n (x)}

= ln (5) \cdot 5^{x s i n (x)} (sin (x) + x cos (x))