derivative of ((5^x/(x^5))^{sin(x^5)})

解

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

(\frac{5 ^{x}}{x ^{5}})^{s i n (x^{5})} (5 x^{4} cos (x^{5}) (ln (5) x - 5 ln (x)) + sin (x^{5}) (ln (5) - \frac{5}{x}))

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

\frac{d}{d x} (sin (x^{5}) ln (\frac{5 ^{x}}{x ^{5}})) = cos (x^{5}) \cdot 5 x^{4} (ln (5) x - ln (x^{5})) + (ln (5) - \frac{5}{x}) sin (x^{5})

= e^{s i n (x^{5}) l n (\frac{5 ^{x}}{x ^{5}})} (cos (x^{5}) \cdot 5 x^{4} (ln (5) x - ln (x^{5})) + (ln (5) - \frac{5}{x}) sin (x^{5}))

簡素化

e^{s i n (x^{5}) l n (\frac{5 ^{x}}{x ^{5}})} (cos (x^{5}) \cdot 5 x^{4} (ln (5) x - ln (x^{5})) + (ln (5) - \frac{5}{x}) sin (x^{5})) : (\frac{5 ^{x}}{x ^{5}})^{s i n (x^{5})} (5 x^{4} cos (x^{5}) (ln (5) x - 5 ln (x)) + sin (x^{5}) (ln (5) - \frac{5}{x}))

= (\frac{5 ^{x}}{x ^{5}})^{s i n (x^{5})} (5 x^{4} cos (x^{5}) (ln (5) x - 5 ln (x)) + sin (x^{5}) (ln (5) - \frac{5}{x}))