(\partial)/(\partial x)(x^{-1}y^2+xy^2+5xy)

解

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 5 x y)

- \frac{y ^{2}}{x ^{2}} + y^{2} + 5 y

解答ステップ

\frac{\partial}{\partial x} (x^{- 1} y^{2} + x y^{2} + 5 x y)

y

を定数として扱う

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (x^{- 1} y^{2}) + \frac{\partial}{\partial x} (x y^{2}) + \frac{\partial}{\partial x} (5 x y)

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

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

\frac{\partial}{\partial x} (5 x y) = 5 y

= - \frac{y ^{2}}{x ^{2}} + y^{2} + 5 y