derivative f(x)=(5+x^4)^{3x}

解

導関数

f (x) = (5 + x^{4})^{3 x}

3 (ln (5 + x^{4}) + \frac{4 x ^{4}}{5 + x ^{4}}) (5 + x^{4})^{3 x}

解答ステップ

\frac{d}{d x} ((5 + x^{4})^{3 x})

指数の規則を適用する:

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

(5 + x^{4})^{3 x} = e^{3 x l n (5 + x^{4})}

= \frac{d}{d x} (e^{3 x l n (5 + x^{4})})

連鎖法則を適用:

e^{3 x l n (5 + x^{4})} \frac{d}{d x} (3 x ln (5 + x^{4}))

= e^{3 x l n (5 + x^{4})} \frac{d}{d x} (3 x ln (5 + x^{4}))

\frac{d}{d x} (3 x ln (5 + x^{4})) = 3 (ln (5 + x^{4}) + \frac{4 x ^{4}}{5 + x ^{4}})

= e^{3 x l n (5 + x^{4})} \cdot 3 (ln (5 + x^{4}) + \frac{4 x ^{4}}{5 + x ^{4}})

e^{3 x l n (5 + x^{4})} = (5 + x^{4})^{3 x}

= 3 (ln (5 + x^{4}) + \frac{4 x ^{4}}{5 + x ^{4}}) (5 + x^{4})^{3 x}