derivative of x^{3/5}sqrt(x^3+2x-6)

解答

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

\frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

求解步骤

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

使用微分乘法定则:

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

f = x^{\frac{3}{5}}, g = x^{3} + 2 x - 6

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

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

\frac{d}{d x} (x^{3} + 2 x - 6) = \frac{3 x ^{2} + 2}{2 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

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

化简

\frac{3}{5 x ^{\frac{2}{5}}} x^{3} + 2 x - 6 + \frac{3 x ^{2} + 2}{2 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}} x^{\frac{3}{5}} : \frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

= \frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}