derivative of (x(x+10)/((x+5)^2))

解

\frac{d}{d x} (\frac{x ( x + 10 )}{( x + 5 ) ^{2}})

\frac{50}{( x + 5 ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{x ( x + 10 )}{( x + 5 ) ^{2}})

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 ( x + 10 ) ) ( x + 5 ) ^{2} - \frac{d}{d x} ( ( x + 5 ) ^{2} ) x ( x + 10 )}{( ( x + 5 ) ^{2} ) ^{2}}

\frac{d}{d x} (x (x + 10)) = 2 x + 10

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

= \frac{( 2 x + 10 ) ( x + 5 ) ^{2} - 2 ( x + 5 ) x ( x + 10 )}{( ( x + 5 ) ^{2} ) ^{2}}

簡素化

\frac{( 2 x + 10 ) ( x + 5 ) ^{2} - 2 ( x + 5 ) x ( x + 10 )}{( ( x + 5 ) ^{2} ) ^{2}} : \frac{50}{( x + 5 ) ^{3}}

= \frac{50}{( x + 5 ) ^{3}}