(\partial)/(\partial x)(7e^{xy}ln(x^2+y^2))

解

\frac{\partial}{\partial x} (7 e^{x y} ln (x^{2} + y^{2}))

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

解答ステップ

\frac{\partial}{\partial x} (7 e^{x y} ln (x^{2} + y^{2}))

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 7 \frac{\partial}{\partial x} (e^{x y} ln (x^{2} + y^{2}))

積の法則を適用:

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

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

= 7 (\frac{\partial}{\partial x} (e^{x y}) ln (x^{2} + y^{2}) + \frac{\partial}{\partial x} (ln (x^{2} + y^{2})) e^{x y})

\frac{\partial}{\partial x} (e^{x y}) = e^{x y} y

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

= 7 (e^{x y} y ln (x^{2} + y^{2}) + \frac{2 x}{x ^{2} + y ^{2}} e^{x y})

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 7 (e^{x y} y ln (x^{2} + y^{2}) + \frac{2 e ^{x y} x}{x ^{2} + y ^{2}})