derivative f(x)=x^{5/11}(x-5)^2

解答

导数

f (x) = x^{\frac{5}{11}} (x - 5)^{2}

\frac{27 x ^{2} - 160 x + 125}{11 x ^{\frac{6}{11}}}

求解步骤

\frac{d}{d x} (x^{\frac{5}{11}} (x - 5)^{2})

使用微分乘法定则:

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

f = x^{\frac{5}{11}}, g = (x - 5)^{2}

= \frac{d}{d x} (x^{\frac{5}{11}}) (x - 5)^{2} + \frac{d}{d x} ((x - 5)^{2}) x^{\frac{5}{11}}

\frac{d}{d x} (x^{\frac{5}{11}}) = \frac{5}{11 x ^{\frac{6}{11}}}

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

= \frac{5}{11 x ^{\frac{6}{11}}} (x - 5)^{2} + 2 (x - 5) x^{\frac{5}{11}}

化简

\frac{5}{11 x ^{\frac{6}{11}}} (x - 5)^{2} + 2 (x - 5) x^{\frac{5}{11}} : \frac{27 x ^{2} - 160 x + 125}{11 x ^{\frac{6}{11}}}

= \frac{27 x ^{2} - 160 x + 125}{11 x ^{\frac{6}{11}}}