derivative y=sqrt(x)(x^4+1)

解

導関数

y = x (x^{4} + 1)

\frac{9 x ^{4} + 1}{2 x}

解答ステップ

\frac{d}{d x} (x (x^{4} + 1))

積の法則を適用:

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

f = x, g = x^{4} + 1

= \frac{d}{d x} (x) (x^{4} + 1) + \frac{d}{d x} (x^{4} + 1) x

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

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

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

簡素化

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

= \frac{9 x ^{4} + 1}{2 x}