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

Lösung

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

3 x^{2} + 10 x - \frac{1}{x ^{2}} + 2

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

x \to 0 lim (\frac{ln ( e ^{x} - x )}{x})

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