derivative of (4x^2+7x^5/(sqrt(x)))

解

\frac{d}{d x} (\frac{4 x ^{2} + 7 x ^{5}}{x})

\frac{3 x ( 7 x ^{3} + 4 )}{2} + 21 x^{\frac{7}{2}}

解答ステップ

\frac{d}{d x} (\frac{4 x ^{2} + 7 x ^{5}}{x})

簡素化

\frac{4 x ^{2} + 7 x ^{5}}{x} : x^{\frac{3}{2}} (7 x^{3} + 4)

= \frac{d}{d x} (x^{\frac{3}{2}} (7 x^{3} + 4))

積の法則を適用:

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

f = x^{\frac{3}{2}}, g = 7 x^{3} + 4

= \frac{d}{d x} (x^{\frac{3}{2}}) (7 x^{3} + 4) + \frac{d}{d x} (7 x^{3} + 4) x^{\frac{3}{2}}

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

\frac{d}{d x} (7 x^{3} + 4) = 21 x^{2}

= \frac{3 x}{2} (7 x^{3} + 4) + 21 x^{2} x^{\frac{3}{2}}

簡素化

\frac{3 x}{2} (7 x^{3} + 4) + 21 x^{2} x^{\frac{3}{2}} : \frac{3 x ( 7 x ^{3} + 4 )}{2} + 21 x^{\frac{7}{2}}

= \frac{3 x ( 7 x ^{3} + 4 )}{2} + 21 x^{\frac{7}{2}}