(f(x)(x)^{g(x)})'

解

(f (x) (x)^{g (x)})^{'}

f' (x) x^{g (x)} + x^{g (x)} (ln (x) g' (x) + \frac{g ( x )}{x}) f (x)

解答ステップ

(f (x) (x)^{g (x)})^{'}

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = f (x), g = x^{g (x)}

= f^{^{'}} (x) x^{g (x)} + (x^{g (x)})^{^{'}} f (x)

(x^{g (x)})^{'} = x^{g (x)} (ln (x) g^{'} (x) + \frac{g ( x )}{x})

= f^{^{'}} (x) x^{g (x)} + x^{g (x)} (ln (x) g^{^{'}} (x) + \frac{g ( x )}{x}) f (x)