derivative of \sqrt[4]{x}^{ln(x})

解

\frac{d}{d x} (4 x^{l n (x)})

\frac{e ^{\frac{l n ^{2} ( x )}{4}} ln ( x )}{2 x}

解答ステップ

\frac{d}{d x} ((4 x)^{l n (x)})

指数の規則を適用する:

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

(4 x)^{l n (x)} = e^{l n (x) l n (4 x)}

= \frac{d}{d x} (e^{l n (x) l n (4 x)})

連鎖法則を適用:

e^{l n (x) l n (4 x)} \frac{d}{d x} (ln (x) ln (4 x))

= e^{l n (x) l n (4 x)} \frac{d}{d x} (ln (x) ln (4 x))

\frac{d}{d x} (ln (x) ln (4 x)) = \frac{ln ( x ^{2} )}{4 x}

= e^{l n (x) l n (4 x)} \frac{ln ( x ^{2} )}{4 x}

簡素化

e^{l n (x) l n (4 x)} \frac{ln ( x ^{2} )}{4 x} : \frac{e ^{\frac{l n ^{2} ( x )}{4}} ln ( x )}{2 x}

= \frac{e ^{\frac{l n ^{2} ( x )}{4}} ln ( x )}{2 x}