derivative of x^4y^5+2xmy(x^2-4y^3a)

解

\frac{d}{d x} (x^{4} y (x)^{5} + 2 x m y (x) (x)^{2} - 4 y (x)^{3} a)

4 x^{3} y (x)^{5} + 5 y (x)^{4} \frac{d}{d x} (y (x)) x^{4} + 2 m (3 x^{2} y (x) + \frac{d}{d x} (y (x)) x^{3}) - 12 a y (x)^{2} \frac{d}{d x} (y (x))

解答ステップ

\frac{d}{d x} (x^{4} y (x)^{5} + 2 x m y (x) (x)^{2} - 4 y (x)^{3} a)

簡素化

x^{4} y (x)^{5} + 2 x m y (x) (x)^{2} - 4 y (x)^{3} a : x^{4} y (x)^{5} + 2 m x^{3} y (x) - 4 a y (x)^{3}

= \frac{d}{d x} (x^{4} y (x)^{5} + 2 m x^{3} y (x) - 4 a y (x)^{3})

和/差の法則を適用:

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

= \frac{d}{d x} (x^{4} y (x)^{5}) + \frac{d}{d x} (2 m x^{3} y (x)) - \frac{d}{d x} (4 a y (x)^{3})

\frac{d}{d x} (x^{4} y (x)^{5}) = 4 x^{3} y (x)^{5} + 5 y (x)^{4} \frac{d}{d x} (y (x)) x^{4}

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

\frac{d}{d x} (4 a y (x)^{3}) = 12 a y (x)^{2} \frac{d}{d x} (y (x))

= 4 x^{3} y (x)^{5} + 5 y (x)^{4} \frac{d}{d x} (y (x)) x^{4} + 2 m (3 x^{2} y (x) + \frac{d}{d x} (y (x)) x^{3}) - 12 a y (x)^{2} \frac{d}{d x} (y (x))