derivative of (1+x^2(1+sqrt(x)))

解

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

\frac{5 x ^{2} + 4 x x + 1}{2 x}

解答ステップ

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

積の法則を適用:

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

f = 1 + x^{2}, g = 1 + x

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

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

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

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

簡素化

2 x (1 + x) + \frac{1}{2 x} (1 + x^{2}) : \frac{5 x ^{2} + 4 x x + 1}{2 x}

= \frac{5 x ^{2} + 4 x x + 1}{2 x}