derivative of 5/3 x^2y^{-2}

解

\frac{d}{d x} (\frac{5}{3} x^{2} y^{- 2})

\frac{5}{3} (\frac{2 x}{y ^{2}} - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}})

解答ステップ

\frac{d}{d x} (\frac{5}{3} x^{2} y^{- 2})

定数を除去:

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

= \frac{5}{3} \frac{d}{d x} (x^{2} y^{- 2})

積の法則を適用:

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

f = x^{2}, g = y^{- 2}

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

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

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

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

簡素化

\frac{5}{3} (2 x y^{- 2} + (- \frac{2 \frac{d y}{d x}}{y ^{3}}) x^{2}) : \frac{5}{3} (\frac{2 x}{y ^{2}} - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}})

= \frac{5}{3} (\frac{2 x}{y ^{2}} - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}})