derivative of 1/2 (tan(x-x))

解

\frac{d}{d x} (\frac{1}{2} (tan (x) - x))

\frac{1}{2} tan^{2} (x)

解答ステップ

\frac{d}{d x} (\frac{1}{2} (tan (x) - x))

定数を除去:

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

= \frac{1}{2} \frac{d}{d x} (tan (x) - x)

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{1}{2} (\frac{d}{d x} (tan (x)) - \frac{d x}{d x})

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

\frac{d x}{d x} = 1

= \frac{1}{2} (sec^{2} (x) - 1)

ピタゴラスの公式を使用する:

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

sec^{2} (x) - 1 = tan^{2} (x)

= \frac{1}{2} tan^{2} (x)