derivative x(x^2+4)^{0.5}

解

導関数

x (x^{2} + 4)^{0.5}

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

解答ステップ

\frac{d}{d x} (x (x^{2} + 4)^{0.5})

積の法則を適用:

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

f = x, g = (x^{2} + 4)^{0.5}

= \frac{d x}{d x} (x^{2} + 4)^{0.5} + \frac{d}{d x} ((x^{2} + 4)^{0.5}) x

\frac{d x}{d x} = 1

\frac{d}{d x} ((x^{2} + 4)^{0.5}) = \frac{x}{( x ^{2} + 4 ) ^{\frac{1}{2}}}

= 1 \cdot (x^{2} + 4)^{0.5} + \frac{x}{( x ^{2} + 4 ) ^{\frac{1}{2}}} x

簡素化

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

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