derivative of (x+1/x (x-1/x))

Lösung

\frac{d}{d x} ((x + \frac{1}{x}) (x - \frac{1}{x}))

2 x + \frac{2}{x ^{3}}

Schritte zur Lösung

\frac{d}{d x} ((x + \frac{1}{x}) (x - \frac{1}{x}))

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

= 2 x + \frac{2}{x ^{3}}

\int e^{2 x} + sin (- x) d x

d er i v a t i v e y = x sin (\frac{2}{x})

t^{2} y^{^{''}} - 2 t y^{^{'}} + 2 y = t ln (t)

t an g e n t x^{2} + x y + y^{2} = 7, (2, 1)

\frac{d}{d x} (4 x^{5} - 3 x^{m} + 2 x^{n} + 5 x^{2} - x - 6)