derivative f(x)=5^{sin(x)}

解

導関数

f (x) = 5^{s i n (x)}

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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