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

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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