derivative y= x/(sqrt(x^2+4))

解

導関数

y = \frac{x}{x ^{2} + 4}

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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