(\partial)/(\partial y)(ln(sqrt(x^2+y^2)))

解

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

\frac{y}{x ^{2} + y ^{2}}

解答ステップ

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

x

を定数として扱う

連鎖法則を適用:

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

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

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

= \frac{1}{x ^{2} + y ^{2}} \cdot \frac{y}{x ^{2} + y ^{2}}

簡素化

\frac{1}{x ^{2} + y ^{2}} \cdot \frac{y}{x ^{2} + y ^{2}} : \frac{y}{x ^{2} + y ^{2}}

= \frac{y}{x ^{2} + y ^{2}}