derivative of (z(z^2x^2))

解

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

2 x z^{3} + 3 z^{2} \frac{d z}{d x} x^{2}

解答ステップ

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

簡素化

(z) (z^{2} x^{2}) : x^{2} z^{3}

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

積の法則を適用:

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

f = x^{2}, g = z^{3}

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

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

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

= 2 x z^{3} + 3 z^{2} \frac{d z}{d x} x^{2}