(\partial)/(\partial x)(3x^2(5x+7y))

解

\frac{\partial}{\partial x} (3 x^{2} (5 x + 7 y))

3 (15 x^{2} + 14 x y)

解答ステップ

\frac{\partial}{\partial x} (3 x^{2} (5 x + 7 y))

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{\partial}{\partial x} (x^{2} (5 x + 7 y))

積の法則を適用:

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

f = x^{2}, g = 5 x + 7 y

= 3 (\frac{\partial}{\partial x} (x^{2}) (5 x + 7 y) + \frac{\partial}{\partial x} (5 x + 7 y) x^{2})

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

\frac{\partial}{\partial x} (5 x + 7 y) = 5

= 3 (2 x (5 x + 7 y) + 5 x^{2})

拡張

2 x (5 x + 7 y) + 5 x^{2} : 15 x^{2} + 14 x y

= 3 (15 x^{2} + 14 x y)