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

解

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

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

解答ステップ

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

積の法則を適用:

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

f = (x - 1)^{2}, g = sin (\frac{1}{x - 1})

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

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

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

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

簡素化

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

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