derivative of ln(sqrt(x^2+y^2))

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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