derivative of sqrt(x)*(x^5+x^4+3x+2)

解

\frac{d}{d x} (x \cdot (x^{5} + x^{4} + 3 x + 2))

\frac{11 x ^{5} + 9 x ^{4} + 9 x + 2}{2 x}

解答ステップ

\frac{d}{d x} (x (x^{5} + x^{4} + 3 x + 2))

積の法則を適用:

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

f = x, g = x^{5} + x^{4} + 3 x + 2

= \frac{d}{d x} (x) (x^{5} + x^{4} + 3 x + 2) + \frac{d}{d x} (x^{5} + x^{4} + 3 x + 2) x

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

\frac{d}{d x} (x^{5} + x^{4} + 3 x + 2) = 5 x^{4} + 4 x^{3} + 3

= \frac{1}{2 x} (x^{5} + x^{4} + 3 x + 2) + (5 x^{4} + 4 x^{3} + 3) x

簡素化

\frac{1}{2 x} (x^{5} + x^{4} + 3 x + 2) + (5 x^{4} + 4 x^{3} + 3) x : \frac{11 x ^{5} + 9 x ^{4} + 9 x + 2}{2 x}

= \frac{11 x ^{5} + 9 x ^{4} + 9 x + 2}{2 x}