derivative of 5xcos(x^2)

Lösung

\frac{d}{d x} (5 x cos (x^{2}))

5 (cos (x^{2}) - 2 x^{2} sin (x^{2}))

Schritte zur Lösung

\frac{d}{d x} (5 x cos (x^{2}))

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Produktregel an:

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

f = x, g = cos (x^{2})

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

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

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

= 5 (1 \cdot cos (x^{2}) + (- sin (x^{2}) \cdot 2 x) x)

Vereinfache

5 (1 \cdot cos (x^{2}) + (- sin (x^{2}) \cdot 2 x) x) : 5 (cos (x^{2}) - 2 x^{2} sin (x^{2}))

= 5 (cos (x^{2}) - 2 x^{2} sin (x^{2}))

x \to + 0 lim (\frac{5}{x ^{2} + x} - \frac{5}{x})

d er i v a t i v e ln (\frac{x - 1}{x + 1})

x \to 3 lim (\frac{sin ( x )}{x})

\int \frac{72}{1 - cos ( 9 x )} d x

t an g e n t \frac{( 4 + x )}{( x - 2 )}, at x = 8