derivative of (4x^4-7x/(x^3-8))

解

\frac{d}{d x} (\frac{4 x ^{4} - 7 x}{x ^{3} - 8})

\frac{4 x ^{6} - 114 x ^{3} + 56}{( x ^{3} - 8 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{4 x ^{4} - 7 x}{x ^{3} - 8})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 4 x ^{4} - 7 x ) ( x ^{3} - 8 ) - \frac{d}{d x} ( x ^{3} - 8 ) ( 4 x ^{4} - 7 x )}{( x ^{3} - 8 ) ^{2}}

\frac{d}{d x} (4 x^{4} - 7 x) = 16 x^{3} - 7

\frac{d}{d x} (x^{3} - 8) = 3 x^{2}

= \frac{( 16 x ^{3} - 7 ) ( x ^{3} - 8 ) - 3 x ^{2} ( 4 x ^{4} - 7 x )}{( x ^{3} - 8 ) ^{2}}

拡張

(16 x^{3} - 7) (x^{3} - 8) - 3 x^{2} (4 x^{4} - 7 x) : 4 x^{6} - 114 x^{3} + 56

= \frac{4 x ^{6} - 114 x ^{3} + 56}{( x ^{3} - 8 ) ^{2}}