derivative of 2tan(x+2tan^3(x))

解

\frac{d}{d x} (2 tan (x) + 2 tan^{3} (x))

2 sec^{2} (x) (1 + 3 tan^{2} (x))

解答ステップ

\frac{d}{d x} (2 tan (x) + 2 tan^{3} (x))

簡素化

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

= \frac{d}{d x} (2 sec^{3} (x) sin (x))

定数を除去:

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

= 2 \frac{d}{d x} (sec^{3} (x) sin (x))

積の法則を適用:

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

f = sec^{3} (x), g = sin (x)

= 2 (\frac{d}{d x} (sec^{3} (x)) sin (x) + \frac{d}{d x} (sin (x)) sec^{3} (x))

\frac{d}{d x} (sec^{3} (x)) = 3 sec^{3} (x) tan (x)

\frac{d}{d x} (sin (x)) = cos (x)

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

簡素化

2 (3 sec^{3} (x) tan (x) sin (x) + cos (x) sec^{3} (x)) : 2 sec^{2} (x) (1 + 3 tan^{2} (x))

= 2 sec^{2} (x) (1 + 3 tan^{2} (x))