derivative of (x^2+4/(4x^3-5x))

解

\frac{d}{d x} (\frac{x ^{2} + 4}{4 x ^{3} - 5 x})

\frac{- 4 x ^{4} - 53 x ^{2} + 20}{( 4 x ^{3} - 5 x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} + 4}{4 x ^{3} - 5 x})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( x ^{2} + 4 ) ( 4 x ^{3} - 5 x ) - \frac{d}{d x} ( 4 x ^{3} - 5 x ) ( x ^{2} + 4 )}{( 4 x ^{3} - 5 x ) ^{2}}

\frac{d}{d x} (x^{2} + 4) = 2 x

\frac{d}{d x} (4 x^{3} - 5 x) = 12 x^{2} - 5

= \frac{2 x ( 4 x ^{3} - 5 x ) - ( 12 x ^{2} - 5 ) ( x ^{2} + 4 )}{( 4 x ^{3} - 5 x ) ^{2}}

拡張

2 x (4 x^{3} - 5 x) - (12 x^{2} - 5) (x^{2} + 4) : - 4 x^{4} - 53 x^{2} + 20

= \frac{- 4 x ^{4} - 53 x ^{2} + 20}{( 4 x ^{3} - 5 x ) ^{2}}