derivative f(x)=\sqrt[x^4]{7-9x}

解

導関数

f (x) = x^{4} 7 - 9 x

(- \frac{4 ln ( 7 - 9 x )}{x ^{5}} - \frac{9}{x ^{4} ( 7 - 9 x )}) (7 - 9 x)^{\frac{1}{x ^{4}}}

解答ステップ

\frac{d}{d x} (x^{4} 7 - 9 x)

指数の規則を適用する:

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

x^{4} 7 - 9 x = e^{\frac{1}{x ^{4}} l n (7 - 9 x)}

= \frac{d}{d x} (e^{\frac{1}{x ^{4}} l n (7 - 9 x)})

連鎖法則を適用:

e^{\frac{1}{x ^{4}} l n (7 - 9 x)} \frac{d}{d x} (\frac{1}{x ^{4}} ln (7 - 9 x))

= e^{\frac{1}{x ^{4}} l n (7 - 9 x)} \frac{d}{d x} (\frac{1}{x ^{4}} ln (7 - 9 x))

\frac{d}{d x} (\frac{1}{x ^{4}} ln (7 - 9 x)) = - \frac{4 ln ( 7 - 9 x )}{x ^{5}} - \frac{9}{x ^{4} ( 7 - 9 x )}

= e^{\frac{1}{x ^{4}} l n (7 - 9 x)} (- \frac{4 ln ( 7 - 9 x )}{x ^{5}} - \frac{9}{x ^{4} ( 7 - 9 x )})

e^{\frac{1}{x ^{4}} l n (7 - 9 x)} = (7 - 9 x)^{\frac{1}{x ^{4}}}

= (- \frac{4 ln ( 7 - 9 x )}{x ^{5}} - \frac{9}{x ^{4} ( 7 - 9 x )}) (7 - 9 x)^{\frac{1}{x ^{4}}}