derivative of sqrt((4+8/x ^2+(2+x)^2))

解

\frac{d}{d x} (4 + \frac{8}{x})^{2} + (2 + x)^{2}

\frac{x ^{3} - 32}{x ^{2} x ^{2} + 16}

解答ステップ

\frac{d}{d x} (4 + \frac{8}{x})^{2} + (2 + x)^{2}

連鎖法則を適用:

\frac{1}{2 ( 4 + \frac{8}{x} ) ^{2} + ( 2 + x ) ^{2}} \frac{d}{d x} ((4 + \frac{8}{x})^{2} + (2 + x)^{2})

= \frac{1}{2 ( 4 + \frac{8}{x} ) ^{2} + ( 2 + x ) ^{2}} \frac{d}{d x} ((4 + \frac{8}{x})^{2} + (2 + x)^{2})

\frac{d}{d x} ((4 + \frac{8}{x})^{2} + (2 + x)^{2}) = 2 x - \frac{64}{x ^{2}} - \frac{128}{x ^{3}} + 4

= \frac{1}{2 ( 4 + \frac{8}{x} ) ^{2} + ( 2 + x ) ^{2}} (2 x - \frac{64}{x ^{2}} - \frac{128}{x ^{3}} + 4)

簡素化

\frac{1}{2 ( 4 + \frac{8}{x} ) ^{2} + ( 2 + x ) ^{2}} (2 x - \frac{64}{x ^{2}} - \frac{128}{x ^{3}} + 4) : \frac{x ^{3} - 32}{x ^{2} x ^{2} + 16}

= \frac{x ^{3} - 32}{x ^{2} x ^{2} + 16}