derivative of \sqrt[4]{x^2+y^2}

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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