derivative of (x+1/x (5x^2-1/(x^2)))

Lösung

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

15 x^{2} + \frac{1}{x ^{2}} + \frac{3}{x ^{4}} + 5

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

\frac{d}{d x} (5 x^{2} - \frac{1}{x ^{2}}) = 10 x + \frac{2}{x ^{3}}

= (1 - \frac{1}{x ^{2}}) (5 x^{2} - \frac{1}{x ^{2}}) + (10 x + \frac{2}{x ^{3}}) (x + \frac{1}{x})

Vereinfache

(1 - \frac{1}{x ^{2}}) (5 x^{2} - \frac{1}{x ^{2}}) + (10 x + \frac{2}{x ^{3}}) (x + \frac{1}{x}) : 15 x^{2} + \frac{1}{x ^{2}} + \frac{3}{x ^{4}} + 5

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

d er i v a t i v e f (x) = - 3 3 x \cdot sec (2 x)

d er i v a t i v e \frac{9 x - 9}{- x + 7}

t an g e n t y = 3 x^{2} - 7 x + 4

\int sec (y) d y

t an g e n t \frac{x ^{2} + 1}{x + e ^{x} + 2}