derivative of ((sqrt(x^2-3x-4))/x)

解

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

\frac{3 x + 8}{2 x ^{2} x ^{2} - 3 x - 4}

解答ステップ

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

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

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

\frac{d x}{d x} = 1

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

簡素化

\frac{\frac{2 x - 3}{2 x ^{2} - 3 x - 4} x - 1 \cdot x ^{2} - 3 x - 4}{x ^{2}} : \frac{3 x + 8}{2 x ^{2} x ^{2} - 3 x - 4}

= \frac{3 x + 8}{2 x ^{2} x ^{2} - 3 x - 4}