derivative of (3+sinh(x)/(1+coth^2(x)))

解

\frac{d}{d x} (\frac{3 + sinh ( x )}{1 + coth ^{2} ( x )})

\frac{cosh ( x ) ( 1 + coth ^{2} ( x ) ) + 2 csch ^{2} ( x ) coth ( x ) ( 3 + sinh ( x ) )}{( 1 + coth ^{2} ( x ) ) ^{2}}

解答ステップ

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

\frac{d}{d x} (3 + sinh (x)) = cosh (x)

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

= \frac{cosh ( x ) ( 1 + coth ^{2} ( x ) ) - ( - 2 csch ^{2} ( x ) coth ( x ) ) ( 3 + sinh ( x ) )}{( 1 + coth ^{2} ( x ) ) ^{2}}

簡素化

= \frac{cosh ( x ) ( 1 + coth ^{2} ( x ) ) + 2 csch ^{2} ( x ) coth ( x ) ( 3 + sinh ( x ) )}{( 1 + coth ^{2} ( x ) ) ^{2}}