derivative of (x+1/(sqrt({f)(x)+4)})

解

\frac{d}{d x} (\frac{x + 1}{f ( x ) + 4})

\frac{2 ( f ( x ) + 4 ) - \frac{d}{d x} ( f ( x ) ) ( x + 1 )}{2 ( f ( x ) + 4 ) f ( x ) + 4}

解答ステップ

\frac{d}{d x} (\frac{x + 1}{f ( x ) + 4})

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 + 1 ) f ( x ) + 4 - \frac{d}{d x} ( f ( x ) + 4 ) ( x + 1 )}{( f ( x ) + 4 ) ^{2}}

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

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

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

簡素化

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

= \frac{2 ( f ( x ) + 4 ) - \frac{d}{d x} ( f ( x ) ) ( x + 1 )}{2 ( f ( x ) + 4 ) f ( x ) + 4}