derivative of a^{{f}(x})

解

\frac{d}{d x} (a^{f (x)})

a^{f (x)} ln (a) \frac{d}{d x} (f (x))

解答ステップ

\frac{d}{d x} (a^{f (x)})

指数の規則を適用する:

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

a^{f (x)} = e^{f (x) l n (a)}

= \frac{d}{d x} (e^{f (x) l n (a)})

連鎖法則を適用:

e^{f (x) l n (a)} \frac{d}{d x} (f (x) ln (a))

= e^{f (x) l n (a)} \frac{d}{d x} (f (x) ln (a))

\frac{d}{d x} (f (x) ln (a)) = ln (a) \frac{d}{d x} (f (x))

= e^{f (x) l n (a)} ln (a) \frac{d}{d x} (f (x))

e^{f (x) l n (a)} = a^{f (x)}

= a^{f (x)} ln (a) \frac{d}{d x} (f (x))