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

解

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

2 x ln (x^{2} + y^{2}) + \frac{2 x ^{3}}{x ^{2} + y ^{2}}

解答ステップ

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

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

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

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

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

\frac{2 x}{x ^{2} + y ^{2}} x^{2} = \frac{2 x ^{3}}{x ^{2} + y ^{2}}

= 2 x ln (x^{2} + y^{2}) + \frac{2 x ^{3}}{x ^{2} + y ^{2}}