derivative of (15x^7+27x^5/(3x))

解

\frac{d}{d x} (\frac{15 x ^{7} + 27 x ^{5}}{3 x})

30 x^{5} + 36 x^{3}

解答ステップ

\frac{d}{d x} (\frac{15 x ^{7} + 27 x ^{5}}{3 x})

簡素化

\frac{15 x ^{7} + 27 x ^{5}}{3 x} : x^{4} (5 x^{2} + 9)

= \frac{d}{d x} (x^{4} (5 x^{2} + 9))

積の法則を適用:

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

f = x^{4}, g = 5 x^{2} + 9

= \frac{d}{d x} (x^{4}) (5 x^{2} + 9) + \frac{d}{d x} (5 x^{2} + 9) x^{4}

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

\frac{d}{d x} (5 x^{2} + 9) = 10 x

= 4 x^{3} (5 x^{2} + 9) + 10 x x^{4}

簡素化

4 x^{3} (5 x^{2} + 9) + 10 x x^{4} : 30 x^{5} + 36 x^{3}

= 30 x^{5} + 36 x^{3}