derivative of \sqrt[3]{x+sqrt(x^5)}

解答

\frac{2 + 5 x ^{\frac{3}{2}}}{6 ( x + x ^{2} x ) ^{\frac{2}{3}}}

求解步骤

使用连锁律:

\frac{1}{3 ( x + x ^{5} ) ^{\frac{2}{3}}} \frac{d}{d x} (x + x^{5})

= \frac{1}{3 ( x + x ^{5} ) ^{\frac{2}{3}}} \frac{d}{d x} (x + x^{5})

\frac{d}{d x} (x + x^{5}) = 1 + \frac{5 x ^{\frac{3}{2}}}{2}

= \frac{1}{3 ( x + x ^{5} ) ^{\frac{2}{3}}} (1 + \frac{5 x ^{\frac{3}{2}}}{2})

化简

\frac{1}{3 ( x + x ^{5} ) ^{\frac{2}{3}}} (1 + \frac{5 x ^{\frac{3}{2}}}{2}) : \frac{2 + 5 x ^{\frac{3}{2}}}{6 ( x + x ^{2} x ) ^{\frac{2}{3}}}

= \frac{2 + 5 x ^{\frac{3}{2}}}{6 ( x + x ^{2} x ) ^{\frac{2}{3}}}