derivative of (y^3/(x^2))

解

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

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

解答ステップ

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

\frac{3 y ^{2} \frac{d y}{d x} x ^{2} - 2 x y ^{3}}{( x ^{2} ) ^{2}} : \frac{y ^{2} ( 3 x \frac{d y}{d x} - 2 y )}{x ^{3}}

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