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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

(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}

Vereinfache

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})

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