derivative of (sqrt(x^2+y-1)/(x+1))

解

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

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

解答ステップ

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

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

\frac{d}{d x} (x^{2} + y - 1) = \frac{2 x + \frac{d y}{d x}}{2 x ^{2} + y - 1}

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

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

簡素化

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

= \frac{2 x + x \frac{d y}{d x} + \frac{d y}{d x} - 2 y + 2}{2 ( x + 1 ) ^{2} x ^{2} + y - 1}