(\partial)/(\partial x)(sin(1/(x^2+y^2)))

解

\frac{\partial}{\partial x} (sin (\frac{1}{x ^{2} + y ^{2}}))

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

解答ステップ

\frac{\partial}{\partial x} (sin (\frac{1}{x ^{2} + y ^{2}}))

y

を定数として扱う

連鎖法則を適用:

cos (\frac{1}{x ^{2} + y ^{2}}) \frac{\partial}{\partial x} (\frac{1}{x ^{2} + y ^{2}})

= cos (\frac{1}{x ^{2} + y ^{2}}) \frac{\partial}{\partial x} (\frac{1}{x ^{2} + y ^{2}})

\frac{\partial}{\partial x} (\frac{1}{x ^{2} + y ^{2}}) = - \frac{2 x}{( x ^{2} + y ^{2} ) ^{2}}

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

簡素化

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

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