derivative of 3x(18x^4+(13/(x+1)))

解

\frac{d}{d x} (3 x (18 x^{4} + \frac{13}{x + 1}))

\frac{3 ( 90 x ^{6} + 180 x ^{5} + 90 x ^{4} + 13 )}{( x + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (3 x (18 x^{4} + \frac{13}{x + 1}))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d x} (x (18 x^{4} + \frac{13}{x + 1}))

積の法則を適用:

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

f = x, g = 18 x^{4} + \frac{13}{x + 1}

= 3 (\frac{d x}{d x} (18 x^{4} + \frac{13}{x + 1}) + \frac{d}{d x} (18 x^{4} + \frac{13}{x + 1}) x)

\frac{d x}{d x} = 1

\frac{d}{d x} (18 x^{4} + \frac{13}{x + 1}) = 72 x^{3} - \frac{13}{( x + 1 ) ^{2}}

= 3 (1 \cdot (18 x^{4} + \frac{13}{x + 1}) + (72 x^{3} - \frac{13}{( x + 1 ) ^{2}}) x)

簡素化

3 (1 \cdot (18 x^{4} + \frac{13}{x + 1}) + (72 x^{3} - \frac{13}{( x + 1 ) ^{2}}) x) : \frac{3 ( 90 x ^{6} + 180 x ^{5} + 90 x ^{4} + 13 )}{( x + 1 ) ^{2}}

= \frac{3 ( 90 x ^{6} + 180 x ^{5} + 90 x ^{4} + 13 )}{( x + 1 ) ^{2}}