d/(d{x)}(({z})(4{z}^3+3{x}{z}^2-{x}{y}{z}-2{x}{y}^2+7))

解

\frac{d}{d x} ((z) (4 z^{3} + 3 x z^{2} - x y z - 2 x y^{2} + 7))

z (3 z^{2} - y z - 2 y^{2})

解答ステップ

\frac{d}{d x} ((z) (4 z^{3} + 3 x z^{2} - x y z - 2 x y^{2} + 7))

定数を除去:

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

= z \frac{d}{d x} (4 z^{3} + 3 x z^{2} - x y z - 2 x y^{2} + 7)

和/差の法則を適用:

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

= z (\frac{d}{d x} (4 z^{3}) + \frac{d}{d x} (3 x z^{2}) - \frac{d}{d x} (x y z) - \frac{d}{d x} (2 x y^{2}) + \frac{d}{d x} (7))

\frac{d}{d x} (4 z^{3}) = 0

\frac{d}{d x} (3 x z^{2}) = 3 z^{2}

\frac{d}{d x} (x y z) = y z

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

\frac{d}{d x} (7) = 0

= z (0 + 3 z^{2} - y z - 2 y^{2} + 0)

簡素化

= z (3 z^{2} - y z - 2 y^{2})