derivative of (1-cosh(x)/(1+cosh(x)))

解

\frac{d}{d x} (\frac{1 - cosh ( x )}{1 + cosh ( x )})

- \frac{2 sinh ( x )}{( 1 + cosh ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1 - cosh ( x )}{1 + cosh ( x )})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d x} (1 - cosh (x)) = - sinh (x)

\frac{d}{d x} (1 + cosh (x)) = sinh (x)

= \frac{( - sinh ( x ) ) ( 1 + cosh ( x ) ) - sinh ( x ) ( 1 - cosh ( x ) )}{( 1 + cosh ( x ) ) ^{2}}

簡素化

\frac{( - sinh ( x ) ) ( 1 + cosh ( x ) ) - sinh ( x ) ( 1 - cosh ( x ) )}{( 1 + cosh ( x ) ) ^{2}} : - \frac{2 sinh ( x )}{( 1 + cosh ( x ) ) ^{2}}

= - \frac{2 sinh ( x )}{( 1 + cosh ( x ) ) ^{2}}