derivative of 1/2 (x-1^3sin(1/(x-1)))

解

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

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

解答ステップ

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

定数を除去:

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

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

積の法則を適用:

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

f = (x - 1)^{3}, g = sin (\frac{1}{x - 1})

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

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

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

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

簡素化

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

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