(\partial)/(\partial x)(x^2+y^2+(12-x-y)^2)

解

\frac{\partial}{\partial x} (x^{2} + y^{2} + (12 - x - y)^{2})

4 x + 2 y - 24

解答ステップ

\frac{\partial}{\partial x} (x^{2} + y^{2} + (12 - x - y)^{2})

簡素化

x^{2} + y^{2} + (12 - x - y)^{2} : 2 x^{2} + 2 x y - 24 x + 2 y^{2} - 24 y + 144

= \frac{\partial}{\partial x} (2 x^{2} + 2 x y - 24 x + 2 y^{2} - 24 y + 144)

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} (2 x^{2}) + \frac{\partial}{\partial x} (2 x y) - \frac{\partial}{\partial x} (24 x) + \frac{\partial}{\partial x} (2 y^{2}) - \frac{\partial}{\partial x} (24 y) + \frac{\partial}{\partial x} (144)

\frac{\partial}{\partial x} (2 x^{2}) = 4 x

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

\frac{\partial}{\partial x} (24 x) = 24

\frac{\partial}{\partial x} (2 y^{2}) = 0

\frac{\partial}{\partial x} (24 y) = 0

\frac{\partial}{\partial x} (144) = 0

= 4 x + 2 y - 24 + 0 - 0 + 0

簡素化

= 4 x + 2 y - 24