derivative of (x^4+7x^3+14x^2/(x+3))

解

\frac{d}{d x} (\frac{x ^{4} + 7 x ^{3} + 14 x ^{2}}{x + 3})

\frac{3 x ^{4} + 26 x ^{3} + 77 x ^{2} + 84 x}{( x + 3 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{4} + 7 x ^{3} + 14 x ^{2}}{x + 3})

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 ^{4} + 7 x ^{3} + 14 x ^{2} ) ( x + 3 ) - \frac{d}{d x} ( x + 3 ) ( x ^{4} + 7 x ^{3} + 14 x ^{2} )}{( x + 3 ) ^{2}}

\frac{d}{d x} (x^{4} + 7 x^{3} + 14 x^{2}) = 4 x^{3} + 21 x^{2} + 28 x

\frac{d}{d x} (x + 3) = 1

= \frac{( 4 x ^{3} + 21 x ^{2} + 28 x ) ( x + 3 ) - 1 \cdot ( x ^{4} + 7 x ^{3} + 14 x ^{2} )}{( x + 3 ) ^{2}}

簡素化

\frac{( 4 x ^{3} + 21 x ^{2} + 28 x ) ( x + 3 ) - 1 \cdot ( x ^{4} + 7 x ^{3} + 14 x ^{2} )}{( x + 3 ) ^{2}} : \frac{3 x ^{4} + 26 x ^{3} + 77 x ^{2} + 84 x}{( x + 3 ) ^{2}}

= \frac{3 x ^{4} + 26 x ^{3} + 77 x ^{2} + 84 x}{( x + 3 ) ^{2}}