(\partial)/(\partial x)((F(x,y))/(V(x,y))x-(F(x,y))/(V(x,y))y-ky^n)

解

\frac{\partial}{\partial x} (\frac{F ( x , y )}{V ( x , y )} x - \frac{F ( x , y )}{V ( x , y )} y - k y^{n})

(\frac{\frac{\partial}{\partial x} ( F ( x , y ) ) V ( x , y ) - \frac{\partial}{\partial x} ( V ( x , y ) ) F ( x , y )}{V ( x , y ) ^{2}} x + 1 \cdot \frac{F ( x , y )}{V ( x , y )}) - y \frac{\frac{\partial}{\partial x} ( F ( x , y ) ) V ( x , y ) - \frac{\partial}{\partial x} ( V ( x , y ) ) F ( x , y )}{V ( x , y ) ^{2}} - 0

解答ステップ

\frac{\partial}{\partial x} (\frac{F ( x , y )}{V ( x , y )} x - \frac{F ( x , y )}{V ( x , y )} y - k y^{n})

k, n, y

を定数として扱う

和/差の法則を適用:

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

= \frac{\partial}{\partial x} (\frac{F ( x , y )}{V ( x , y )} x) - \frac{\partial}{\partial x} (\frac{F ( x , y )}{V ( x , y )} y) - \frac{\partial}{\partial x} (k y^{n})

\frac{\partial}{\partial x} (\frac{F ( x , y )}{V ( x , y )} x) = \frac{\frac{\partial}{\partial x} ( F ( x , y ) ) V ( x , y ) - \frac{\partial}{\partial x} ( V ( x , y ) ) F ( x , y )}{V ( x , y ) ^{2}} x + 1 \cdot \frac{F ( x , y )}{V ( x , y )}

\frac{\partial}{\partial x} (\frac{F ( x , y )}{V ( x , y )} y) = y \frac{\frac{\partial}{\partial x} ( F ( x , y ) ) V ( x , y ) - \frac{\partial}{\partial x} ( V ( x , y ) ) F ( x , y )}{V ( x , y ) ^{2}}

\frac{\partial}{\partial x} (k y^{n}) = 0

= (\frac{\frac{\partial}{\partial x} ( F ( x , y ) ) V ( x , y ) - \frac{\partial}{\partial x} ( V ( x , y ) ) F ( x , y )}{V ( x , y ) ^{2}} x + 1 \cdot \frac{F ( x , y )}{V ( x , y )}) - y \frac{\frac{\partial}{\partial x} ( F ( x , y ) ) V ( x , y ) - \frac{\partial}{\partial x} ( V ( x , y ) ) F ( x , y )}{V ( x , y ) ^{2}} - 0