derivative of (3x^2-1/(2x+5))

解

\frac{d}{d x} (\frac{3 x ^{2} - 1}{2 x + 5})

\frac{6 x ^{2} + 30 x + 2}{( 2 x + 5 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{3 x ^{2} - 1}{2 x + 5})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 3 x ^{2} - 1 ) ( 2 x + 5 ) - \frac{d}{d x} ( 2 x + 5 ) ( 3 x ^{2} - 1 )}{( 2 x + 5 ) ^{2}}

\frac{d}{d x} (3 x^{2} - 1) = 6 x

\frac{d}{d x} (2 x + 5) = 2

= \frac{6 x ( 2 x + 5 ) - 2 ( 3 x ^{2} - 1 )}{( 2 x + 5 ) ^{2}}

拡張

6 x (2 x + 5) - 2 (3 x^{2} - 1) : 6 x^{2} + 30 x + 2

= \frac{6 x ^{2} + 30 x + 2}{( 2 x + 5 ) ^{2}}