derivative of (x-5xsqrt(x)/(sqrt(x)))

解

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

\frac{1 - 5 x}{2 x} - \frac{5}{2}

解答ステップ

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

簡素化

\frac{x - 5 x x}{x} : x (1 - 5 x)

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

積の法則を適用:

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

f = x, g = 1 - 5 x

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

\frac{d}{d x} (x) = \frac{1}{2 x}

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

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

簡素化

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

= \frac{1 - 5 x}{2 x} - \frac{5}{2}