derivative of x^{1/2}y

解

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

\frac{y + 2 x \frac{d y}{d x}}{2 x ^{\frac{1}{2}}}

解答ステップ

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

積の法則を適用:

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

f = x^{\frac{1}{2}}, g = y

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

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

= \frac{1}{2 x ^{\frac{1}{2}}} y + \frac{d y}{d x} x^{\frac{1}{2}}

簡素化

\frac{1}{2 x ^{\frac{1}{2}}} y + \frac{d y}{d x} x^{\frac{1}{2}} : \frac{y + 2 x \frac{d y}{d x}}{2 x ^{\frac{1}{2}}}

= \frac{y + 2 x \frac{d y}{d x}}{2 x ^{\frac{1}{2}}}