derivative of xcos(1/x)

Lösung

\frac{d}{d x} (x cos (\frac{1}{x}))

cos (\frac{1}{x}) + \frac{sin ( \frac{1}{x} )}{x}

Schritte zur Lösung

\frac{d}{d x} (x cos (\frac{1}{x}))

Wende die Produktregel an:

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

f = x, g = cos (\frac{1}{x})

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

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

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

= 1 \cdot cos (\frac{1}{x}) + \frac{sin ( \frac{1}{x} )}{x ^{2}} x

Vereinfache

1 \cdot cos (\frac{1}{x}) + \frac{sin ( \frac{1}{x} )}{x ^{2}} x : cos (\frac{1}{x}) + \frac{sin ( \frac{1}{x} )}{x}

= cos (\frac{1}{x}) + \frac{sin ( \frac{1}{x} )}{x}

x \to - 1 lim ((x + 2 x^{3})^{4})

\int \frac{x ^{5}}{1 - 2 x ^{2}} d x

\int 1 + 4 r^{2} r d r

\int \frac{3}{e ^{x}} d x

d er i v a t i v e f (x) = 2^{x^{2}}