derivative ((2x^2+3x))/(4x^2)

解

導関数

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

- \frac{3}{4 x ^{2}}

解答ステップ

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

定数を除去:

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

= \frac{1}{4} \frac{d}{d x} (\frac{2 x ^{2} + 3 x}{x ^{2}})

商の法則を適用:

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

= \frac{1}{4} \cdot \frac{\frac{d}{d x} ( 2 x ^{2} + 3 x ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( 2 x ^{2} + 3 x )}{( x ^{2} ) ^{2}}

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

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

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

簡素化

\frac{1}{4} \cdot \frac{( 4 x + 3 ) x ^{2} - 2 x ( 2 x ^{2} + 3 x )}{( x ^{2} ) ^{2}} : - \frac{3}{4 x ^{2}}

= - \frac{3}{4 x ^{2}}