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

解答

\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)

求解步骤

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

使用微分乘法定则:

(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)

e x p an d \frac{x}{( x - 4 ) ( x - 2 )}

e x p an d (\frac{16 x - 2}{4 x ^{2} + 20}) (- 4)

e x p an d (1 - \frac{1}{2})^{21}

e x p an d π (- \frac{π}{4} + 1)

\frac{d ^{2}}{d x ^{2}} ((y) (y^{2} + y + 2 x^{3}))