derivative (5sqrt(x)+2)(1/x+3x^5)

解

導関数

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

\frac{5 ( 1 + 3 x ^{6} )}{2 x x} + (- \frac{1}{x ^{2}} + 15 x^{4}) (5 x + 2)

解答ステップ

\frac{d}{d x} ((5 x + 2) (\frac{1}{x} + 3 x^{5}))

積の法則を適用:

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

f = 5 x + 2, g = \frac{1}{x} + 3 x^{5}

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

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

\frac{d}{d x} (\frac{1}{x} + 3 x^{5}) = - \frac{1}{x ^{2}} + 15 x^{4}

= \frac{5}{2 x} (\frac{1}{x} + 3 x^{5}) + (- \frac{1}{x ^{2}} + 15 x^{4}) (5 x + 2)

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

= \frac{5 ( 1 + 3 x ^{6} )}{2 x x} + (- \frac{1}{x ^{2}} + 15 x^{4}) (5 x + 2)