derivative f(x)=x^{7/5}(x+5)

解答

导数

f (x) = x^{\frac{7}{5}} (x + 5)

\frac{12 x ^{\frac{7}{5}} + 35 x ^{\frac{2}{5}}}{5}

求解步骤

\frac{d}{d x} (x^{\frac{7}{5}} (x + 5))

使用微分乘法定则:

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

f = x^{\frac{7}{5}}, g = x + 5

= \frac{d}{d x} (x^{\frac{7}{5}}) (x + 5) + \frac{d}{d x} (x + 5) x^{\frac{7}{5}}

\frac{d}{d x} (x^{\frac{7}{5}}) = \frac{7 x ^{\frac{2}{5}}}{5}

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

= \frac{7 x ^{\frac{2}{5}}}{5} (x + 5) + 1 \cdot x^{\frac{7}{5}}

化简

\frac{7 x ^{\frac{2}{5}}}{5} (x + 5) + 1 \cdot x^{\frac{7}{5}} : \frac{12 x ^{\frac{7}{5}} + 35 x ^{\frac{2}{5}}}{5}

= \frac{12 x ^{\frac{7}{5}} + 35 x ^{\frac{2}{5}}}{5}

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