derivative of \sqrt[3]{y}*\sqrt[4]{x^3}

解

\frac{d}{d x} (3 y \cdot 4 x^{3})

\frac{4 x ^{3} \frac{d y}{d x}}{3 y ^{\frac{2}{3}}} + \frac{3 3 y}{4 x ^{\frac{1}{4}}}

解答ステップ

\frac{d}{d x} (3 y 4 x^{3})

積の法則を適用:

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

f = 3 y, g = 4 x^{3}

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

\frac{d}{d x} (3 y) = \frac{\frac{d y}{d x}}{3 y ^{\frac{2}{3}}}

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

= \frac{\frac{d y}{d x}}{3 y ^{\frac{2}{3}}} 4 x^{3} + \frac{3}{4 x ^{\frac{1}{4}}} 3 y

簡素化

\frac{\frac{d y}{d x}}{3 y ^{\frac{2}{3}}} 4 x^{3} + \frac{3}{4 x ^{\frac{1}{4}}} 3 y : \frac{4 x ^{3} \frac{d y}{d x}}{3 y ^{\frac{2}{3}}} + \frac{3 3 y}{4 x ^{\frac{1}{4}}}

= \frac{4 x ^{3} \frac{d y}{d x}}{3 y ^{\frac{2}{3}}} + \frac{3 3 y}{4 x ^{\frac{1}{4}}}