(\partial)/(\partial x)(xln(x^2+y^2))

解

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

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

解答ステップ

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

y

を定数として扱う

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = ln (x^{2} + y^{2})

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

\frac{\partial}{\partial x} (x) = 1

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

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

簡素化

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

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