derivative of 2x^3+1/(x^2+16x^{7/2}+5)

解

\frac{d}{d x} (2 x^{3} + \frac{1}{x ^{2}} + 16 x^{\frac{7}{2}} + 5)

\frac{2 ( 3 x ^{5} + 28 x ^{\frac{11}{2}} - 1 )}{x ^{3}}

解答ステップ

\frac{d}{d x} (2 x^{3} + \frac{1}{x ^{2}} + 16 x^{\frac{7}{2}} + 5)

簡素化

2 x^{3} + \frac{1}{x ^{2}} + 16 x^{\frac{7}{2}} + 5 : \frac{1 + 2 x ^{5} + 16 x ^{\frac{11}{2}}}{x ^{2}} + 5

= \frac{d}{d x} (\frac{1 + 2 x ^{5} + 16 x ^{\frac{11}{2}}}{x ^{2}} + 5)

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (\frac{1 + 2 x ^{5} + 16 x ^{\frac{11}{2}}}{x ^{2}}) + \frac{d}{d x} (5)

\frac{d}{d x} (\frac{1 + 2 x ^{5} + 16 x ^{\frac{11}{2}}}{x ^{2}}) = \frac{2 ( 3 x ^{5} + 28 x ^{\frac{11}{2}} - 1 )}{x ^{3}}

\frac{d}{d x} (5) = 0

= \frac{2 ( 3 x ^{5} + 28 x ^{\frac{11}{2}} - 1 )}{x ^{3}} + 0

簡素化

= \frac{2 ( 3 x ^{5} + 28 x ^{\frac{11}{2}} - 1 )}{x ^{3}}