(\partial)/(\partial x)(arctan(t/x))

解

\frac{\partial}{\partial x} (arctan (\frac{t}{x}))

- \frac{t}{t ^{2} + x ^{2}}

解答ステップ

\frac{\partial}{\partial x} (arctan (\frac{t}{x}))

t

を定数として扱う

連鎖法則を適用:

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

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

\frac{\partial}{\partial x} (\frac{t}{x}) = - \frac{t}{x ^{2}}

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

簡素化

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

= - \frac{t}{t ^{2} + x ^{2}}