(\partial)/(\partial x)(arctan(y/(x+a)))

解

\frac{\partial}{\partial x} (arctan (\frac{y}{x + a}))

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

解答ステップ

\frac{\partial}{\partial x} (arctan (\frac{y}{x + a}))

a, y

を定数として扱う

連鎖法則を適用:

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

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

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

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

簡素化

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

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