derivative of tan(sqrt(cos^2(x^5)))

解

\frac{d}{d x} (tan (cos^{2} (x^{5})))

- 5 x^{4} sec^{2} (cos (x^{5})) sin (x^{5})

解答ステップ

\frac{d}{d x} (tan (cos^{2} (x^{5})))

連鎖法則を適用:

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

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

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

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

簡素化

sec^{2} (cos^{2} (x^{5})) (- \frac{5 x ^{4} cos ( x ^{5} ) sin ( x ^{5} )}{cos ^{2} ( x ^{5} )}) : - 5 x^{4} sec^{2} (cos (x^{5})) sin (x^{5})

= - 5 x^{4} sec^{2} (cos (x^{5})) sin (x^{5})