(\partial)/(\partial x)(xy(1-x^2-y^2))

解

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

y (- 3 x^{2} - y^{2} + 1)

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

f = x, g = 1 - x^{2} - y^{2}

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

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

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

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

簡素化

y (1 \cdot (1 - x^{2} - y^{2}) + (- 2 x) x) : y (- 3 x^{2} - y^{2} + 1)

= y (- 3 x^{2} - y^{2} + 1)