derivative (x^4-sec(4x^2-2))^{-4}

解

導関数

(x^{4} - sec (4 x^{2} - 2))^{- 4}

- \frac{4 ( 4 x ^{3} - 8 x sec ( 4 x ^{2} - 2 ) tan ( 4 x ^{2} - 2 ) )}{( x ^{4} - sec ( 4 x ^{2} - 2 ) ) ^{5}}

解答ステップ

\frac{d}{d x} ((x^{4} - sec (4 x^{2} - 2))^{- 4})

連鎖法則を適用:

- \frac{4}{( x ^{4} - sec ( 4 x ^{2} - 2 ) ) ^{5}} \frac{d}{d x} (x^{4} - sec (4 x^{2} - 2))

= - \frac{4}{( x ^{4} - sec ( 4 x ^{2} - 2 ) ) ^{5}} \frac{d}{d x} (x^{4} - sec (4 x^{2} - 2))

\frac{d}{d x} (x^{4} - sec (4 x^{2} - 2)) = 4 x^{3} - sec (4 x^{2} - 2) tan (4 x^{2} - 2) \cdot 8 x

= - \frac{4}{( x ^{4} - sec ( 4 x ^{2} - 2 ) ) ^{5}} (4 x^{3} - sec (4 x^{2} - 2) tan (4 x^{2} - 2) \cdot 8 x)

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{4 ( 4 x ^{3} - 8 x sec ( 4 x ^{2} - 2 ) tan ( 4 x ^{2} - 2 ) )}{( x ^{4} - sec ( 4 x ^{2} - 2 ) ) ^{5}}