derivative of x(a+x^2(a-x)^3)

解

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

a^{2} (a - x)^{3} + 4 a x (a - x)^{3} + 3 x^{2} (a - x)^{3} - 3 x (a - x)^{2} (a + x)^{2}

解答ステップ

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

積の法則を適用:

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

f = x, g = (a + x)^{2} (a - x)^{3}

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

\frac{d x}{d x} = 1

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

= 1 \cdot (a + x)^{2} (a - x)^{3} + (2 (a + x) (a - x)^{3} - 3 (a - x)^{2} (a + x)^{2}) x

簡素化

1 \cdot (a + x)^{2} (a - x)^{3} + (2 (a + x) (a - x)^{3} - 3 (a - x)^{2} (a + x)^{2}) x : a^{2} (a - x)^{3} + 4 a x (a - x)^{3} + 3 x^{2} (a - x)^{3} - 3 x (a - x)^{2} (a + x)^{2}

= a^{2} (a - x)^{3} + 4 a x (a - x)^{3} + 3 x^{2} (a - x)^{3} - 3 x (a - x)^{2} (a + x)^{2}