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

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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