derivative of ((1-cos(x)/(1+cos(x)))^3)

解

\frac{d}{d x} ((\frac{1 - cos ( x )}{1 + cos ( x )})^{3})

\frac{6 sin ( x ) ( 1 - cos ( x ) ) ^{2}}{( 1 + cos ( x ) ) ^{4}}

解答ステップ

\frac{d}{d x} ((\frac{1 - cos ( x )}{1 + cos ( x )})^{3})

連鎖法則を適用:

3 (\frac{1 - cos ( x )}{1 + cos ( x )})^{2} \frac{d}{d x} (\frac{1 - cos ( x )}{1 + cos ( x )})

= 3 (\frac{1 - cos ( x )}{1 + cos ( x )})^{2} \frac{d}{d x} (\frac{1 - cos ( x )}{1 + cos ( x )})

\frac{d}{d x} (\frac{1 - cos ( x )}{1 + cos ( x )}) = \frac{2 sin ( x )}{( 1 + cos ( x ) ) ^{2}}

= 3 (\frac{1 - cos ( x )}{1 + cos ( x )})^{2} \frac{2 sin ( x )}{( 1 + cos ( x ) ) ^{2}}

簡素化

3 (\frac{1 - cos ( x )}{1 + cos ( x )})^{2} \frac{2 sin ( x )}{( 1 + cos ( x ) ) ^{2}} : \frac{6 sin ( x ) ( 1 - cos ( x ) ) ^{2}}{( 1 + cos ( x ) ) ^{4}}

= \frac{6 sin ( x ) ( 1 - cos ( x ) ) ^{2}}{( 1 + cos ( x ) ) ^{4}}