derivative of ({f}(x,z,y)(zx^2-xy^2-yz^2))

解

\frac{d}{d x} ((f (x, z, y)) (z (x) x^{2} - x y^{2} - yz (x)^{2}))

\frac{d}{d x} (f (x, z, y)) (z x^{2} - x y^{2} - y z^{2}) + (x^{2} \frac{d z}{d x} + 2 x z - y^{2} - 2 x y \frac{d y}{d x} - \frac{d y}{d x} z^{2} - 2 z \frac{d z}{d x} y) f (x, z, y)

解答ステップ

\frac{d}{d x} ((f (x, z, y)) (z x^{2} - x y^{2} - y z^{2}))

積の法則を適用:

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

f = f (x, z, y), g = z x^{2} - x y^{2} - y z^{2}

= \frac{d}{d x} (f (x, z, y)) (z x^{2} - x y^{2} - y z^{2}) + \frac{d}{d x} (z x^{2} - x y^{2} - y z^{2}) f (x, z, y)

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

= \frac{d}{d x} (f (x, z, y)) (z x^{2} - x y^{2} - y z^{2}) + (x^{2} \frac{d z}{d x} + 2 x z - y^{2} - 2 x y \frac{d y}{d x} - \frac{d y}{d x} z^{2} - 2 z \frac{d z}{d x} y) f (x, z, y)