derivative f(x)=sqrt(x)(3x+x^2)

解

導関数

f (x) = x (3 x + x^{2})

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

解答ステップ

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

積の法則を適用:

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

f = x, g = 3 x + x^{2}

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

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

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

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

簡素化

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

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