derivative of (3x^2+5x/(sqrt(x)))

解

\frac{d}{d x} (\frac{3 x ^{2} + 5 x}{x})

\frac{9 x + 5}{2 x}

解答ステップ

\frac{d}{d x} (\frac{3 x ^{2} + 5 x}{x})

簡素化

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

= \frac{d}{d x} (x (3 x + 5))

積の法則を適用:

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

f = x, g = 3 x + 5

= \frac{d}{d x} (x) (3 x + 5) + \frac{d}{d x} (3 x + 5) x

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

\frac{d}{d x} (3 x + 5) = 3

= \frac{1}{2 x} (3 x + 5) + 3 x

簡素化

\frac{1}{2 x} (3 x + 5) + 3 x : \frac{9 x + 5}{2 x}

= \frac{9 x + 5}{2 x}