derivative of (5/x+x(5/x-x))

解

\frac{d}{d x} ((\frac{5}{x} + x) (\frac{5}{x} - x))

- 2 x - \frac{50}{x ^{3}}

解答ステップ

\frac{d}{d x} ((\frac{5}{x} + x) (\frac{5}{x} - x))

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = \frac{5}{x} + x, g = \frac{5}{x} - x

= \frac{d}{d x} (\frac{5}{x} + x) (\frac{5}{x} - x) + \frac{d}{d x} (\frac{5}{x} - x) (\frac{5}{x} + x)

\frac{d}{d x} (\frac{5}{x} + x) = - \frac{5}{x ^{2}} + 1

\frac{d}{d x} (\frac{5}{x} - x) = - \frac{5}{x ^{2}} - 1

= (- \frac{5}{x ^{2}} + 1) (\frac{5}{x} - x) + (- \frac{5}{x ^{2}} - 1) (\frac{5}{x} + x)

簡素化

(- \frac{5}{x ^{2}} + 1) (\frac{5}{x} - x) + (- \frac{5}{x ^{2}} - 1) (\frac{5}{x} + x) : - 2 x - \frac{50}{x ^{3}}

= - 2 x - \frac{50}{x ^{3}}