derivative of 5sec(5xtan(5x))

解

\frac{d}{d x} (5 sec (5 x) tan (5 x))

5 (5 tan^{2} (5 x) sec (5 x) + 5 sec^{3} (5 x))

解答ステップ

\frac{d}{d x} (5 sec (5 x) tan (5 x))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{d}{d x} (sec (5 x) tan (5 x))

積の法則を適用:

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

f = sec (5 x), g = tan (5 x)

= 5 (\frac{d}{d x} (sec (5 x)) tan (5 x) + \frac{d}{d x} (tan (5 x)) sec (5 x))

\frac{d}{d x} (sec (5 x)) = sec (5 x) tan (5 x) \cdot 5

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

= 5 (sec (5 x) tan (5 x) \cdot 5 tan (5 x) + sec^{2} (5 x) \cdot 5 sec (5 x))

簡素化

5 (sec (5 x) tan (5 x) \cdot 5 tan (5 x) + sec^{2} (5 x) \cdot 5 sec (5 x)) : 5 (5 tan^{2} (5 x) sec (5 x) + 5 sec^{3} (5 x))

= 5 (5 tan^{2} (5 x) sec (5 x) + 5 sec^{3} (5 x))