derivative of sec^2(xtan^2(x))

解

\frac{d}{d x} (sec^{2} (x) tan^{2} (x))

2 tan^{3} (x) sec^{2} (x) + 2 sec^{4} (x) tan (x)

解答ステップ

\frac{d}{d x} (sec^{2} (x) tan^{2} (x))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = sec^{2} (x), g = tan^{2} (x)

= \frac{d}{d x} (sec^{2} (x)) tan^{2} (x) + \frac{d}{d x} (tan^{2} (x)) sec^{2} (x)

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

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

= 2 sec^{2} (x) tan (x) tan^{2} (x) + 2 tan (x) sec^{2} (x) sec^{2} (x)

簡素化

2 sec^{2} (x) tan (x) tan^{2} (x) + 2 tan (x) sec^{2} (x) sec^{2} (x) : 2 tan^{3} (x) sec^{2} (x) + 2 sec^{4} (x) tan (x)

= 2 tan^{3} (x) sec^{2} (x) + 2 sec^{4} (x) tan (x)